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§ 4 Primitive Near-Rings
102
$ 4 PRIMITIVE NEAR-RINGS
This paranraph presents a discussion o f t h e " b u i l d i n q stones, near-rinqs
a r e made o f " ,
Similar t o r i n q theory,
the so-called t h e "atoms"
"primitive near-rinqs".
a r e n o t t h e simple near-
r i n g s as one m i o h t e x p e c t a t a f i r s t g l a n c e . an i m p o r t a n t c o n n e c t i o n (4.47).
g i v e n a n e a r - r i n q N,
f r u i t s (= N-groups)
r e c o q n i z e them b y
we l o o k a t a l l o f i t s
and ask, w h e t h e r t h e r e a r e f a i t h f u l and
' ' e n o u g h s i m p l e " o n e s amonq t h e m . N " p r i m i t i v e on t h i s N-group". p r e c i s e we f i x
however,
The i d e a t o c o n s i d e r p r i m i t i v e
n e a r - r i n g s comes f r o m t h e b i b l e ( " Y o u w i l l their fruits"):
There i s ,
If t h i s i s t h e c a s e , we c a l l
S i n c e "enouqh s i m p l e "
i s not
i t s menninq i n w a n t i n g N-qroups o f t y p e v.
The r e s u l t i n q c o n c e p t i s t h a t o f " v - p r i m i t i v i t y " . We g e t t h e h i e r a r c h y 2 - p r f r n i t i v i t y < T > l - p r i m i t i v i t y < Z ' mitivity,
discuss conditions,
O-pri-
w h i c h f o r c e some o f t h e s e c o n c e p t s
t o c o i n c i d e a n d make a l o t o f w o r k t o w a r d s a d e n s i t y t h e o r e m which i s c o m p a r a t l e t o t h e c e l e b r a t e d one i n r i n o t h e o r y due
t o N . J a c o b s o n . We r e a l l y q e t o n e f o r 2 - p r i m i t i v e n e a r - r i n q s w i t h i d e n t i t y (4.52).
A d d i n q a c h a i n c o n d i t i o n , we a r r i v e a t
a Wedderburn-Artin-like
s t r u c t u r e theorem (4.60).
Before that,
we g e t " b e t t e r a n d b e t t e r " d e n s i t y - l i k e s t r u c t u r e t h e o r e m s f o r 0-,
1- a n d 2 - p r i m i t i v e n e a r - r i n g s .
theorems on v - p r i m i t i v e symmetric v - p r i m i t i v e
I t comes o u t t h a t m a n y
n e a r - r i n q s c a n be d e r i v e d f r o m z e r o -
n e a r - r i n q s w h e r e t h e y a r e much e a s i e r
t o o b t a i n s i n c e t h e s e ones behave more l i k e r i n q s . many p r o o f s c o n c e r n i n g e v e n z e r o - s y m m e t r i c
However,
near-rinqs
differ
t o t a l l y f r o m t h e comparable ones i n r i n g t h e o r y . Anyhow,
t h e " b u i l d i n g stones" mentioned above ( 2 - p r i m i t i v e
n e a r - r i n g s w i t h i d e n t i t y ) a r e shown t o b e d e n s e i n or
Maff(r) ( i f
MGo"k51 ( r ) t M c ( T ) fixed-point-free
No
i s a ring) or i n
( i f No
MGo,{al
i s a non-ring),
automorphism group
where
(r).
AutN 0
HomD(r,r)
(r)
or
Go
i s the
I n particular,
4a General
103
i f Go = ( i d ) , t h e l a t t e r t w o o n e s a r e Mo(r) a n d M ( r ) . F i n a l l y , t h e d e n s i t y p r o p e r t y i s s e e n t o be a k i n d o f a n i n t e r p o l a t i o n p r o p e r t y and a " p u r e l y i n t e r p o l a t i o n - t h e o r e t i c " r e s u l t w i l l be o b t a i n e d . R e c a l l a g a i n ( p . 1 ) t h a t r * = r \ { o ) , a n d so o n .
a 1
G E N E R A L
1 . ) DEFINITIONS A N D E L E M E N T A R Y RESULTS 4.1
CONVENTION I n a l l w h a t f o l l o w s , w w i l l be a n y n u m b e r unless otherwise specified.
E I O , ,Zl ~ 4.2
DEFINITION
( a ) 1.4 i s c a l l e d u - p r i m i t i v e on
Nr:
<->
Nr
is f a i t h f u l
o f type w.
y:
( b ) ri i s v - p r i m i t i v e : <=> 3 N r E N N i s w - p r i m i t i v e on ( c ) I A N i s c a l l e d a u - p r i m i t i v e i d e a l o f N : <-> N/I w - p r i mi t i ve . 4.3
Nr. is
T h e n the following
PROPOSITION L e t ' I be an i d e a l o f N . conditions are equivalent: ( a ) I i s w-primitive. (b)
3 NrENq:
I =
(o:r)
( c ) 3 L s$, N : I = ( L : N ) Proof.
( a ) ->
Nr is
A A
o f type w.
L i s v-modular.
( b ) : I i s w - p r i m i t i v e -> N/I i s u - p r i m i t i v e N / I r -> Nr ( a s i n 3 . 1 4 ( b ) ) i s o f t y p e w a n d
o n some I = (o:r).
( b ) -> ( c ) : L e t r be = Ny p I o l . ( 0 : ~ )= : L . Then 'L L i s m o d u l a r . By 3 . 4 ( e ) , N / L r , so L i s w-modular. Finally, I = ( o : r ) = (o:N/L) = (L:N). nN
( c ) => ( a ) : T a k e N / L = : r . T h e n ( a s above) I = (L:N) = ( o : r ) .
Nr
i s o f type w and
8 4 PRIMITIVE NEAR-RINGS
104
4.4
COROLLARY
The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t :
(a) N i s w-primitive. (b) (c)
4.5
I01
3
i s a w-primitive ideal.
L 3% N :
t w-modular A (L:N)
IOI.
=
REMARKS ( a ) Observe t h a t ( c ) i n 4.3 and 4 . 4 q i v e " i n t r i n s i c " characterizations o f primitivity
-
t h a t w i l l be
f o r i t e n a b l e s one t o r e c o g n i z e
extremely helpful,
p r i m i t i v i t y " w i t h i n N". (b) I f
N i s v - p r i m i t i v e on r t h e n IA N
if
i s a w-primitive
(c) 2-primitivity
Nv
( d ) The n e a r - r i n g s (e)
ideal then
r
Io?
1 4
N.
and
i m p l i e s 1 - p r i m i t i v i t y and t h i s i n t u r n
implies 0-primitivity n e a r - r i nqs.
N $. f 0 1 ,
( a l w a y s o n t h e same g r o u p ) .
o f 3 .8 a r e e x a m p l e s o f w - p r i m i t i v e
(on Z4).
If N i s v-primitive
on
r
then
M(r)
N 4
(1.48).
( f ) See 95 o f B e t s c h ( 3 ) f o r a d i s c u s s i o n o f t h e s p a c e s o f w-primitive 4.6
PROPOSITION L e t i d e n t i t y e.
( v = 1,2)
ideals
of
NEWo.
N contain either a l e f t or a riqht
Then
(a) Every u - p r i m i t i v e i d e a l I o f N i s modular.
(b) I f e i s a l e f t i d e n t i t y o f N t h e n N i s 1 - p r i m i t i v e iff N i s 2-primitive
(and i n t h i s case e i s a two-
sided identity). Proof.
( a ) I f e i s a l e f t i d e n t i t y i n N t h e n (because
N/I
i s w - p r i m i t i v e o n some
of
N/I
by 3.4(c).
So
N/Ir) e + I
\f nEN: e n
f
i s an i d e n t i t y
n e E n {mod I ) .
If e i s a r i g h t i d e n t i t y , t h e a s s e r t i o n i s t r i v i a l . (b) L e t N be I - p r i m i t i v e on 3.4(c),
Nr
e i s a two-sided
i s unitary.
Nr.
By
i d e n t i t y f o r N.
Flow a p p l y 3 . 7 ( c )
By 3 . 4 ( b ) ,
and 3 . 1 9 ( a ) .
105
4a General
4.7
PROPOSITION L e t FI be s i m p l e a n d N i s v - p r i m i t i v e on r. P r o o f . (o:r) 4 N , so the c o n t r a d i c t i o n
4.8
Nr
(o:r) = to1 Nr = { o l ) .
b e o f t y p e w . Then
(for
(o:r) = N
gives
PROPOSITION ( B e t s c h ( 3 ) ) . L e t t h e r i n g N be w - p r i m i t i v e on r . T h e n N i s a p r i m i t i v e r i n g on t h e N-module r ( ( N . J a c o b s o n ) , p. 4 ) .
Proof, I f
r
= Ny
abelian. If nEN:
then
% r -N N/(o:y)
nl+(o:$),
(r,+)
is
n 2 + ( o : y ) ~ N / ( o : y ) then
n(nl+(o:y)+n2+(o:y))
= n ( n , t ( o : y ) ) + n ( n 2 + ( o: y ) )
v
and
.
= nnlt(o:y)+nn2t(o:y)
=
Hence tf y l , y 2 ~ r nEN: n(y1ty2) nyltny2, and N r i s a ( r i n g - ) module. Each N-submodule o f N r i s a n i d e a l , s o = l o ) o r = r . F i n a l l y , Nr p { o ) by a s s u m p t i o n , s o Nr i s i r r e d u c i b l e a n d N i s p r i m i t i v e on r . 4.9
C O R O L L A R Y (Ramakotaiah ( 1 ) ) . I f N i s commutative and u - p r i m i t i v e then N i s a f i e l d .
LEX~(N):
(L:N) = {OI. L 9 N , s i n c e P r o o f , BY 4 . 4 ( ~ ) , 3 N i s c o m m u t a t i v e . By 3 . 2 5 , (L:N) is the greatest i d e a l i n L, s o L = (01 a n d N c o n t a i n s a r i g h t i d e n t i t y . By 1 . 1 0 7 ( c ) , N i s a r i n g , h e n c e a p r i m i t i v e r i n g by 4 . 8 a n d by ( N . J a c o b s o n ) , p . 7 a f i e l d . In ( 3 ) , R a m a k o t a i a h s h o w s t h a t i f I q N = N o a n d I €g0(t\1) t h e n I is a 0 - p r i m i t i v e ideal. Near-rinas N with a f a i t h f u l , s i m p l e , n o n t r i v i a l N-group a r e c a l l e d ? - p r i m i t i v e a n d a r e s t u d i e d in H a r t n e y ( 4 ) , M e l d r u m ( 7 ) , ( 1 3 ) . S e e a l s o B e i d l e m a n (7),(8),(9). H o l c o m b e blalker ( 1 ) s t u d y 3 - p r i m i t i v e n e a r - r i n q s PI, w h i c h m e a n s t h a t N h a s a f a i t h f u l N-group o f t y p e 3 ( s e e t h e l a s t l i n e s o f p. 80).
8 4 PRIMITIVE NEAR-RINGS
106
2 . ) T H E CENTRALIZER
4 . 1 0 DEFINITION E n d N ( r ) = H m N ( r , r ) =: C N ( r ) = : C i s c a l l e d t h e c e n t r a l i z e r o f Nr ( c f . K a a r l i ( 2 1 , R a m a k o t a i a h ( 3 ) ) .
A u t N ( r j =: G o .
A u t N ( r ) =: G N ( r ) =: G;
0
Go:
if
=
~ E C ;
otherwise 4 . 1 1 REMARKS ( a ) (C,.)
C Q = {o} ( b ) ~ E <=> (c) If 4 . 1 2 NOTATION
(ci,o)
.
N 4 M C N ( r ) (l-1 5
is f a i t h f u l then
NT
If
fn:
nEN,
l i k e w i s e .:G
a n d (Go,") are q r o u p s , ( " g r o u p s w i t h z e r o " ) a r e rnonoids.
i s a monoid, a n d (Go,,")
(GO,")
;
r
-+
'I
Y
-c
ny
;
FN(r):=
M&9.
ffnlnENl =: F .
4 . 1 3 PROPOSITION ( M l i t z ( 3 ) ) .
(a) If
= ImcM(r)I
(b)
tl
hECN(r):
(c) If Proof.
tl
CN(r)= fom) = : MF(r).
Nr i s m o n o g e n i c t h e n
Q =
r
(a) If YET:
WfEF: mof = h/n = id.
then
CN(r) = { i d } .
hECN(r)
and
fnEFN(r)
then
(hofn)(y) = h(ny) = nh(y) = ( f n o h ) ( y ) ;
so
hEMF(r).
sMF(r). I f y 1 , y 2 & r = Ny a n d y 1 = n l y A y 2 = n 2 y . Then
C o n v e r s e l y , l e t f be
n&N
then
f(nv1) =
3
nl,n2EN:
(fof,)(vl)
=
(fnof)(vl)
= nf(Y1)
and
107
4a General
Hence
feCN(r).
(b)V
heCN(r)
noc:R: h ( n o ) = n h ( o ) = n o .
( c ) f o l l o w s from ( b ) . 1.14 N O T A T I O N
eo:
=
e o ( N r ) := { y c r
el:
=
el(Nr):
Ny = NO =
Rl.
ri.
= C Y E ~ NY =
1 . 1 5 REMARKS ( B e t s c h ( 6 ) ) .
(a)
oEeo,
e l $. 0 ( c ) eon e l (b)
P 0.
so
eo
<->
Nr i s m o n o q e n i c . n $. r .
= 0 <->
r
( d ) Nr i s s t r o n q l y m o n o g e n i c -> ( e ) Nr
eo
i s u n i t a r y ->
and
=> y = l y c R
( f ) G ( B o ) = Bo g r o u p s on
=
A G(B1)
eo
and
(for
= R
w =
= Bow
el,
y e e 0 ->
Ny
R ==>
nocR => Nu = NnocNo = R =I>
so G i n d u c e s p e r m u t a t i o n
el, (if
WE^^)
+ 0)
el
on
el.
The n e x t p r o p o s i t i o n i s a " S c h u r - t y p e lemma". 4 . 1 6 P R O P O S I T I O N ( B e t s c h (S), M l i t z ( 3 ) ) . ( a ) Nr i s s i m p l e (hEC A 2
(b)
n
A
yEB1:
= { o } ->
h(y)EOl)
Nr i s N - s i m p l e
->
C = Colu M o n N ( r )
he6.
->
C = Epi,,,(r,Q)u E p i N ( r , r )
EpiN(r,n) = {el!). (c)
Nr i s N o - s i m p l e
=>
C = G0
and
.
(if
NEW,,
84 PRIMITIVE NEAR-RINGS
108
Proof. ( a ) follows from the f a c t t h a t h s C : Ker h so e i t h e r Ker h = Col ( t h e n h s M o n N ( r ) ) o r Ker h = r ( t h e n h = 0 ) . We may assume t h a t I f hEC A ] yeel: h(y)sel then h 9 6 , so h s M o n N ( r ) . Now h ( r ) = h(Ny) = Nh(y) = r . ( b ) V hsC: Im h = r .
Im h sN
r,
so e i t h e r
Im h = R
SIN
r
r,
+ Iol.
or
( c ) f o l l o w s from ( b ) . i n which We a r e mainly i n t e r e s t e d i n t h e c a s e t h a t C = G o , e v e r y n o n - z e r o N-endomorphism o f r i s an N-automorphism. 4.17 PROPOSITION (Betsch ( 6 ) ) .
( a ) G i s f i x e d - p o i n t - f r e e ( 1 . 4 ( b ) ) on e l . (b)
Nr i s s i m p l e t h e n
If
P r o o f . ( a ) Assume t h a t f o r
'd
6Er
3
= ny = 6 .
( b ) ->:
Then
nEN:
So
6 = ny.
C = Go <-> qEG
Then
and
1
A
yeel
r : r iN A. ~ ( y )= y .
g ( 6 ) = g(ny) = ng(y> =
g = id.
Assume t h a t h i s an N-isomorphism r + b c N h s C = G 0 c {6lu A u t N ( r ) , a c o n t r a d i c t i o n .
r.
I f hEC, h 6 then Ker h 9 r . s o Ker h = I o l . ?r T h e r e f o r e h i s a monomorphism and r Irn h . So Im h = r , a n d h s A u t N ( r ) . <=-:
-
4.18 C O R O L L A R Y ( B e t s c h ( 6 ) ) . I f Nr i s o f t y p e 1 o r i f i s s i m p l e a n d f i n i t e t h e n C = Go.
Nr
P r o o f . I f N r i s of t y p e 1 t h e n Nr i s s i m p l e . Assume t h a t h i s a n N-isomorphism r * A < N r. R e p r e s e n t r a s r = Ny and c a l l h ( y ) = : 6 . N6 = Nh(y) = h(Ny) = = h(r) = A. I f & E e l t h e n N6 = r . s o r = A , a c o n t r a d i c t i o n . I f beeo t h e n N6 = R = I o l , s o A = { o l a n d t h e r e f o r e r = C o l , which a q a i n i s a c o n t r a d i c t i o n .
109
4a General
Now a p p l y 4.15(d) a n d 4 . 1 7 ( b ) .
Nr
If 4.19 NOTATION
i s s i m p l e and f i n i t e ,
For
apply 4.17(bj.
y , 6 ~ ~ we r define (o:y)N
<->
6:
y
=
(0:a)
0 ?r
y
4.20
%
No
;
Go(y) = G o ( 6 ) .
6 : <->
REMARKS ( B e t s c h ( 5 ) ) .
(a)
%,?
( b ) The e q u i v a l e n c e c l a s s e s o f of (c)
r.
are equivalence relations i n
r.
on
Go
y,6Er:
=> (0:y)
%
y%6 -> NO
=
(for
p 6
(o:q(y))
The r e a s o n f o r d z f i n i n g
% %,%
?
are exactly the o r b i t s
= (0:s)
NO
via
3
%
y%6 ->
No
0
=> y a ) .
i n s t e a d o f N stems
f r o m 4 . 1 3 ( c ) : i n t h e f r e q u e n t case t h a t R = o t h e r w i s e be t h e a l l - r e l a t i o n i n a n y c a s e . 4.21 P R O P O S I T I O N (Betsch ( 6 ) ) . % t h e n ?, a n d % c o i n c i d e o n Proof.
If
nly
y%6
If
h : 'I ny
+
+
r
n6
= ( o : 6 ) =>
N = No
nl,n2~N n16
i s well defined.
t o be an N-automorphism,
Now
would
el.
= n 2 y ==> nl-n2E(o:y)
Therefore
r, ?
Nr i s u n i t a r y a n d
then f o r a l l
(y,6Ee1)
g ( y ) = 6 ==>
gEGo:
so
h ( y ) = h ( 1 y ) = 1 6 = 6,
= n26.
h turns out
hEG. hence
y
5
6.
84 PRIMITIVE NEAR-RINGS
110
3 . ) INDEPENDENCE A N D D E N S I T Y
An a p p r o p r i a t e frame f o r o u r
n e x t c o n s i d e r a t i o n s i s g i v e n by
4 . 2 2 D E F I N I T I O N ( M l i t t ( 9 ) ) . L e t M be a n a r b i t r a r y s e t a n d t h e s e t of a l l f i n i t e s u b s e t s o f M . A map f(M) r : f ( M ) + INo i s c a l l e d a r a n k map i f
(a) r(0) = 0 (b)
1
(c)
ti F€f!(M)
FE$(M)
mEM:
ti
r(F u E m l )
m,ncM:
= r(F)+a
with
aEI0,l)
[ r ( F u ( m l ) = r ( F W I n 1 ) = r ( F ) ->
r ( F w { m , n I ) = r(F)].
->
F i s then c a l l e d r-independent i f
r(F) = IF[.
4.23 R E M A R K
With r e s p e c t t o r - i n d e p e n d e n c e , S t e i n i t z ' s t h e o r e m i s f u l f i l l e d ( s e e A . Kertesz, "On independent s e t s of
e l e m e n t s i n a l g e b r a " , Acta S c i . M a t h . 260 - 2 6 9 ) . S e e a l s o K a a r l i ( 2 ) .
(Szesed) 21, 1960,
4.24 EXAMPLES
r ( F ) : = I F [ . Then r i s a r a n k f u n c t i o n a n d every ( f i n i t e ) subset i s r-independent.
( a ) Define
( b ) Take a v e c t o r s p a c e M o v e r a f i e l d K .
Set r(F): = = dim L ( F ) ( l i n e a r h u l l ) . r i s a rank function and r-independence i s j u s t l i n e a r independence.
( c ) Take a n N - g r o u p r a n d d e f i n e f o r e a c h
r(@) a s t h e number o f non % - e q u i v a l e n t g e n e r a t o r s ( i . e . r ( Q ) = l @ n 6 1 / Q l ) . Then r i s a r a n k f u n c t i o n a n d #
= {y l y . . . , y n l
i s r-independent
if
oEf(I')
#eel
tr i S j : v l h j . This independence i s c a l l e d %-independence. The same c a n be d o n e f o r 5.
and
111
4a General
In the theory of rings each primitive r i n g R i s isomorphic t o a " d e n s e " s u b r i n g T'? o f a r i n g HomD(I',I') f o r some i r r e d u c i b l e R-module I' a n d w i t h D = HomR(I',I') ( t h e c e n t r a l i z e r ) making Dr i n t o a v e c t o r s p a c e ( s e e ( N . J a c o b s o n ) , p . 26 3 1 ) . D e n s i t y means h e r e ( i n o u r n o t a t i o n ) t h a t SEN ( y l , ...,y s l lin. indep. i n r 1 61 6 5 ~ r 3 FeR j i E : ( l , sl: r(yi) = 6i.
-
...,
,...,
( I t i s c l e a r t h a t o n l y v a l u e s o f i n d e p e n d e n t e l e m e n t s c a n be arbitrarily prescribed.) We a r e g o i n g t o p r o v e s i m i l a r t h e o r e m s f o r n e a r - r i n g s . B u t b e f o r e d o i n g s o we h a v e t o t a k e a l o o k a t t h e d e n s i t y c o n c e p t ( s e e a l s o Adler ( 1 ) and Ramakotaiah-Rao ( I ) ) . 4 . 2 5 N O T A T I O N L e t M be a s u b s e t o f some a t o p o l o g y i n M as i n B e t s c h ( 7 ) :
We i n t r o d u c e
M(T).
I f mcM a n d y c r , d e f i n e S ( m , y ) : and q : = fS(m,y)[mcM A y E r 1 .
= Im'EM(m'(y)
= m(y)l
4 . 2 6 P R O P O S I T I O N ( B e t s c h (7),(11)). (a)
yis
t h e s u b b a s e o f some t o p o l o g y t o p o l o g y " ) on M.
i s dense i n M
( b ) NEM
<->
V
SEIN
tf m e M
w.r.t.
7
yl, . . . , y
7
(the "finite
<==>
s ~ r3
nEN
V
ie(l,
...,$ 1 :
: n ( q ) = m(v+
P r o o f . s t r a i g h t f o r w a r d and hence o m i t t e d . I n a l l t h a t f o l l o w s , " d e n s i t y " means " d e n s i t y w i t h r e s p e c t t o o f 4.26". 4.27 REMARKS ( a ) I f M and N a r e s u b n e a r - r i n g s o f M ( r ) t h e n i t i s easy t o see t h a t No i s d e n s e i n Mo i f f No+Mc(r) i s d e n s e i n Mo+Mc(r). N o t e t h a t N o t M c ( r ) a n d M o + M c ( r ) a r e no n e a r - r i n g s i n g e n e r a l ( s e e 4 . 5 3 ( e ) ) , e x c e p t i n some i m p o r t a n t s p e c i a l c a s e s . ( S e e 4 . 5 4 and 4 . 6 0 . )
112
$ 4 PRIMITIVE NEAR-RINGS
( b ) If N i s d e n s e i n i n Mo.
M then
9 {id)
( c ) Observe t h a t i f H
r
automorphism group of
tf
b
mEMH(r)
= NnMo(r)
i s dense
i s a fixed-point-free
MH(r) r
then
(since
Mo(l')
h(m(o)) = m(o)).
hEH:
If H = { i d )
No:
then
M(r).
MH(r) =
( d ) We w i l l b e m a i n l y i n t e r e s t e d i n n e a r - r i n g s w h i c h a r e d e n s e i n M (r) a n d 77 (I-):= M (r)+Mc(r) G: G: G: ( 4 . 5 2 and o t h e r s ; . 4 . 2 8 T H E O R E M ( R a m a k o t a i a h (2), B e t s c h ( 7 ) ) .
L e t H be a f i x e d -
r.
p o i n t - f r e e g r o u p o f a u t o m o r p h i s m s o f some g r o u p
(a)
W
v
A
(b)
ti
YET*
3
6Er
mEMH(r):
~ ' E P \ H ~ m: ( y ' ) =
tf S E N b
y1 y
0).
., Y ~ E ~ *
. .
,
Hyi
6 1 y . . . y 6 s ~ r 3 mEMH(r) (c) If
H
6
<->
b
SEN
tf
{id),
W
Mc(r)
(d) I f
3
61y
Proof.
W
nEN
c RH(r),
N
v
V SEN
<->
v
E
iE{l,...,sI:
, Hri
Y~,....Y,E~*
Hyi
...,d S € r 3
nEN
Hyj
tj i c { l , . . . , s I :
I n a n y c a s e we may assume t h a t RH(r) = MH(r)
otherwise (a)
aEHy
-fl
haEH:
Define
mEM(r)
by
a = h,(y)
m(y)
= 6
and
0
mEMH(r);
d e t e r m i n e d by t h e c o n d i t i o n s ( v y ' ~ r \ H y : m(y')
9j
= bi.
HH(r) <-> for
i
+j
= 6i.
{id),
for
(since H i s fixed-
haV)
m(a):= I
clearly
n(yi)
n(yi) H
i
for
M(r).
point-free).
A
=+
= 6i.
m(yi)
Hyj
=/=
N i s dense i n
ylY...,ysEr,
j
M H ( r ) <->
... ,s?:
i E { l ,
i
for
Hyj
i s dense i n
NrMH(r)
61y...y6s~r
= 6 A
(m(y)
= 0).
m(y)
aEHY
.
Then a&HY m i s uniquely = 6 A
113
4a General
( b ) D e f i n e maps
mi€MH(r)
b
= o
mi(y')
y'4Hyi:
m: = m l +
Then
with
...+m S
w i l l do t h e j o b .
By ( b ) a n d 4 . 2 6 ( b ) .
<-:
If
Hyi
p
Hyj
If
Hyi
= Hyj,
V
n(yj)
and
= tii
(as i n ( a ) ) .
( c ) =+:
->
mi(yi)
for
i
9 b
mEMH(r)
the result i s clear.
j,
= m(yi)
nEN: n ( y i )
a>
and t h e r e s u l t f o l l o w s a g a i n f r o m
= m(yj)
4.26( b ) . ( d ) +: By 4.27,
I f one
( s a y yl)
yi
= bi-6
V
fulfills
ncEMc(r) Take
= 61.
...,sl.
for
i ~ { 2 ,
ic{l,.,.,s):
Two o r m o r e
take
= 0,
map w h i c h i s c o n s t a n t no(yi)
n(yi)
=
MH(r).
t o be t h e
n0ENo
with n: = notn
Then
C
= tii.
c a n n o t be z e r o .
yi
(R,,(r))o
i s dense i n
No
If all
yi
the
0,
r e s u l t follows from ( c ) . <==: I f
and
yl,
SEN,
... , y S c r * define
61,...,6s~r,
Then
3
n(o) =
nEN 0,
iE{l,
nENo
so by 4 . 2 7 a )
,
Y ~ + ~= :o
..., s + l l :
and b y ( c ) ,
M i s dense i n
4.29 THEOREM ( B e t s c h ( 7 ) ) .
No
i
and
6s+1:
=
= tii.
H 5 Aut(T)
and
r
Then t h e f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t :
(a)
i s discrete i n
RH(r).
(b)
i s discrete i n
MH(r).
( c ) H h a s o n l y f i n i t e l y many o r b i t s o n
( i n t h i s case, H
+ {id}.
( a ) ->
Then
r
t r i v i a l l y hold f o r
i s finite). since
MH(r),
as i n
r. H = {id).
S o we assume t h a t
MH(r)Em0.
(b): Trivial,
0.
Because o f
i s dense i n
4.26.
Proof. Again the r e s u l t s
j
g,,(r).
rev,
Let
n(yi)
for
Hyj
Hyi
MH(r) f
FIH(r).
§4 PRIMITIVE NEAR-RINGS
I14 ( b ) ->
( c ) : Assume t h a t H h a s i q f i n i t e l y many o r b i t s
r.
on
3
Then
If
msMH(r) a n d a n e i g h t o u r h o o d U o f m.
Take
3
SEN
y1
= HyJ
Hyi
assume t h a t
,..., y S ~ F :U
then Hyi
S(m,yi)
+ Hyj
for
S
3
fl
i = l
S(m,yi). So we w i l l
= S(m,yj).
i
p
j.
S i n c e H has i n f i n i t e l y many o r b i t s ,
3
y S t l E r \ ( { o l u Hyl u
\
Then
Then
ml:
=
= m.I
P0
If m ( y S t 1 ) = so
m2
Anyhow,
m2
0
0
3
Hy,).
\y'
eiE:MH(I')
...t e s )
m (el+
and
ml
If m ( Y s t l ) m1
...,s t l l
i t C l ,
Define
...
and
E
fl S(m,yi) i=l
then
ml(Ys+l)
then
m2(yStl) = Y
dp m . U c o n t a i n s an e l e m e n t
= 0
+m
...,s t l l :
= mltestl.
m2:
S
are
jE.Il,
5 U.
=I= m(yst,), ~
+
and
+ 0 ~= m
so
(
~
c a n n o t be
discrete. ( c ) ->
(a):
I f H h a s o n l y f i n i t e l y many o r b i t s o n
t h e n each element o f
MH(r)
and o f
flH(r) i s
u n i q u e l y d e t e r m i n e d b y i t s e f f e c t o n f i n i t e l y many s u i t a b l e elements o f So
7is
discrete.
r.
r
~
~
115
4b 0-primitive near-rings
b l 0-PRIMITIVE N E A R - R I N G S
Now we s h a l l p r o v e a " d e n s i t y - l i k e "
s t r u c t u r e theorem f o r
0 - p r i m i t i v e n e a r - r i n g s , Gle s t a r t w i t h z e r o - s y m m e t r i c o n e s . We may a s ~ u m e ( 1 . 4 6 ) t h a t i f N i s 0 - p r i m i t i v e o n r t h e n N Q M ( T ) . G e n e r a l i z a t i o n s c a n b e f o u n d i n M l i t z (4),(3),(12) a n d K a a r l i ( 6 ) .
4 . 3 0 T H E O R E M ( B e t s c h ( 6 ) ) . L e t NewO be 0 - p r i m i t i v e o n r . I f N i s a r i n g t h e n N i s a p r i m i t i v e r i n o on t h e N-module
r and Jacobson's d e n s i t y theorem i s a p p l i c a b l e . I f N i s a n o n - r i n g t h e n we q e t a k i n d o f a d e n s i t y property:
...,y , ~ r , i c { l , ..., s ? : nyi
(D): \ scIN
3
ncN
Proof.
61,...,6s~r
6i.
N i s a r i n g we o n l y h a v e t o a p p l y 4 . 8 .
If
Now l e t l e t
V
%-indep.
yl,
N
s be
I n t h e t e r m i n o l o g y o f (D),
be a n o n - r i n g . > 1
and f o r
v
Lemma.
t
...,S I : fl
kc{ttl,
Vtc{l,
...,s - 1 1
tc{l,
be t h e statement
i = l
...,s - 1 1 :
(0:Yi)
let
$ (0:Yk)
S(t)
,
S(t).
P r o o f . By i n d u c t i o n o n t . Since f o r
(o:y)
YEel
V
i d e a l o f N,
i,jE{l,
i s a maximal l e f t
...,s l :
(o:yi)~
( o : y . ) = ( o : y . ) => yi%yj -=> i = j . J 1 J Particularly: S(1): k~(2, s l : (o:yl)$
E ( o : ~ . ) =>
v
...,
9(O:Yk). Now assume Then Since
S(t),
t
fl
s 2 3
(O:Yi)$(O:Yk) i=l (o:yk)
and
and
...,sl.
k~ftt2,
(o:Yt+l)$(o:Yk)*
i s maximal,
t
[I
(o:yi)+(o:Yk)
i=1
= (o:Yt+l)+(o:Yk)
= N*
$ 4 PRIMITIVE NEAR-RINGS
116
Since N i s not a r i n q , by 3 . 4 ( i ) , $(o:yk) than S f t t l ) .
t
n
n (o:yttl)+
(o:vi)
i=l
which i s n o t h i n q e l s e
Now r e t u r n t o t h e p r o o f o f 4.28 and l e t yl, . . . , y , , be a s i n (D). A g a l n we u s e i n d u c t i o n o n 61,.,.,6 tc{l,
S
...,s l .
t = 1
If
then
b
Now assume t h a t
b
iE{1,
..., t l :
By t h e lemma,
3
Therefore
Now we t a k e
: nttlyi
t
n
i=l
Since
(o:vi)$(o:y
LYttl
RcL: Lyt+l
= 6i,
nlyl
3
= 61.
ntEN
= 6i.
ntyi
nttl:
nlaN:
tE{l,,..,s-ll
L: =
+ {ol.
Lyttl
3
ylEel
$N
t+l),
hence
(3*4(a)),
= 6ttl-nt6t+l
=
LYttl
r.
.
and g e t \1 i E { l , and t h e p r o o f i s c o m p l e t e . = Ltnt
...,t t l l :
4.31 R E M A R K S
(a)
(D)
i s no " r e a l " d e n s i t y p r o p e r t y s i n c e t h e r e i s no
n e a r - r i n g i n s i g h t i n which f i n i t e topology). ( b ) From
and
(D) i t
N i s dense ( w . r . t t h e
+)
f o l l o w s (Ramakotaiah ( 2 ) ) t h a t ,
if
SEIN
a r e %-independent,
yl,...,yscr
?r
i=l
= W(Yi))). ( c ) The c o n t e n t o f 1Bli
= 1,
(0)
m i g h t be v e r y t h i n :
i f e.g.
(D) i s t r i v i a l . So i t i s n o t t o o s u r p r i s i n g
t h a t t h e c o n v e r s e o f 4.28 does n o t h o l d :
+)
( B e t s c h ) : If o n e c h a n g e s o f k.25 t o Y':={S(m,y)ln&:Mn A y€e,(r)] t h e n o n e g e t s a " r e a l " d e n s i t y t h e o r e m w.r.t. t h e r e s u l t i n g c o a r s e r topology.See also 9.230.
4b 0-primitive near-rings
117
L e t N be t h e n o n - r i n q {fEMo(iZ4)lf(2)E{0,21Af(3) 3f(l)I. In z4, O 1 = { 1 , 3 1 , 1 % 3 %( D ) i s f u l f i l l e d , b u t I0,21 QN Z4, so Z4 i s n o t s i m p l e a n d t h e r e f o r e N i s n o t 0 - p r i m i t i v e on Z 4 .
( d ) ( D ) i s equivalent t o t h e following property: (0’):
3
v
SEN
iEI1,
nEN
yl,. . . , y ,cr,
%-indep.
mE:M(r)
.,.,sI: nyi = m ( y i ) .
Now we t u r n t o a r b i t r a r y n e a r - r i n n s . 4.32 T H E O R E M
( a ) Let N be 0 - p r i m i t i v e o n r. Case 1: Nor -/= { 0 1 . Then N o i s 0 - p r i m i t i v e on so 4.30 i s applicable ( f o r r ) , and NC Case 2 :
5
NO
Mc(r).
Nor
(01.
r,
Then
Mc(r)
N =
and
r
i s a
non-zero simple g r o u p i s 0 - p r i m i t i v e on r a n d ( b ) Conversely, i f e i t h e r N o N, E Mc(r) o r i f N = Mc(r) where r (01 i s s i m p l e t h e n N i s 0 - p r i m i t i v e on Proof.
( a ) Anyhow,
If and
Nor
I01
e
N,
Mc(r).
then NO
No
r.
r
i s 0 - p r i m i t i v e on
i s o f t y p e 0 by 3 . 1 8 ( a )
r
(3.15(a)).
“by f a i t h ” , s o R = r a n d N = N, = Mc(r) by 1 . 5 0 ( b ) . S i n c e Ncr i s s i m p l e i f f r i s s i m p l e , ( a ) i s shown ( o b s e r v e t h a t Ncr Col!). If
Nor
=
(01,
No
= {01
+
( b ) A g a i n b y 3 . 1 8 ( t h i s t i m e by ( b ) ) , i f
N r i s o f t y p e 0 . S i n c e NccMc(r), N o a n d N, ( a n d hence N ) a c t f a i t h f u l l y o n r , s o N i s 0 - p r i m i t i v e o n I’. I f N = Mc(r), r $. Iol and s i m p l e , t h e r e s u l t i s clear.
0 - p r i m i t i v e then
118
4.33
§ 4 PRIMITIVE NEAR-RINGS
R E M A RK
(D) w o u l d
and i n
No
= (0:6)
r,
( i n
if
n o t n e c e s s a r i l y mean t h e same i n would be d e f i n e d by
%
Nr
(o:y)
y%fi:<->
=
il). Cf. 4 . 1 9 .
ideal i s a
4 . 3 4 THEOREM ( R a m a k o t a i a h ( 1 ) ) . E a c h 0 - p r i m i t i v e prime i d e a l
dp N.
P r o o f . L e t I b e a 0 - p r i m i t i v e i d e a l o f N. L e t Nr b e o f t y p e 0 w i t h g e n e r a t o r yo s u c h t h a t I = ( 0 : r ) (4.3). Assume t h a t
Ji $ (o:r),
J1J2
Jir SO
9 N: J1J2tI
= JiNy
(o:r) = I,
=
c Jiyo
+ {ol.
= Jiyo
Jiyo
0
r.
Now
J1 $ I
A 5
A J2$I.
Jir.
Since
By 3 . 4 ( a ) ,
r,
= Jlr =
JlJ2r
a contradiction.
R E MA R K I n 5 . 4 0 we w i l l s e e t h a t t h e c o n v e r s e o f 4 . 3 4
holds i f 4.36
r.
dN
Jiyo
4.35
J1,J2
i ~ { 1 , 2 1 , Jir
For
so
3
N = No
h a s t h e DCCN.
THEOREM ( R a m a k o t a i a h ( 1 ) ) . E v e r y m a x i m a l m o d u l a r i d e a l
I of
Nsno
Proof.
i s a 0 - p r i m i t i v e one.
L e t I b e a m o d u l a r m a x i m a l i d e a l . By 3 . 2 2 ,
I is
c o n t a i n e d i n a modular maximal l e f t i d e a l L . Since (L:N)
i s the largest ideal of N contained i n L
( b y 3.25), we g e t
by 3.21(a). By 4 . 3 ( c ) ,
I?(L:N)
and
(L:N)
By t h e m a x i m a l i t y o f I,
I i s 0-primitive,
i s modular
I = (L:N).
s i n c e b y 3.29 L i s
0-modul a r . By t h e w a y ,
if N i s 0 - p r i m i t i v e on
n o t necessarily simple (K.
r
a n d Ny = : A t h e n A i s
Kaarli).
F o r t h e r e s t o f t h i s s e c t i o n , we g i v e a d e s c r i p t i o n o f a c l a s s o f 0 - p r i m i t i v e near-rinqs which are n o t 1 - p r i m i t i v e . This d i s c u s s i o n i s d u e t o H o l c o m b e (5), w h e r e t h e p r o o f s c a n b e found,
too.
119
4b 0-primitive near-rings
4.37
4.38 DEFINITION I f
:'G
=
EFr, H < A u t ( T ) ,
(r,+)E(a,
t h e t r i p l e (T,B,H)
be the s e t of
r\el
NOTATION I f N r E N q , l e t A : = generators". If A sN r , let (cf. 3.14(a)!).
-
"non-
AutN/(o:A) ( A )
H ( B ) ~ B we , ,-all
c o m p a t i b l e i f a t l e a s t one o f t h e
following conditions i s satisfied:
r.
( a ) B i s no n o r m a l s u b g r o u p o f
3 YEAB 2 (3 h ' c H 3
(b) (c)
(3
A
4.39
BEB
tl
y~r\B
+
hEH : y+B h(y). R E B : y+B = h ' ( y ) ) h
1
Y ' E ~ \ B: h ' ( y ' ) - y ' & B ) .
be 0 - p r i m i t i v e on r , N a n o n - r i n o w i t h NE% i d e n t i t y a n d DCCL, a n d l e t A ( a s i n 4 . 3 7 ) b e an N - s u b g r o u p
THEOREM L e t
of
r
Then
such t h a t
i s not f a i t h f u l , b u t o f type 2.
NA
N i s n o t 1 - p r i m i t i v e on A,
many o r b i t s o n el,
G ( 4 . 1 0 ( b ) ) has f i n i t e l y
i s c o m p a t i b l e and
(T,A,G)
(where A i s a f i n i t e dimensional v e c t o r space o v e r the d i v i s i o n r i n g
GAvta)).
Conversely:
4.40
THEOREM L e t subgroup.
r
Let
b e an a d d i t i v e g r o u p and A be a n o n - z e r o
GA
be a r e g u l a r group o f
automorphisms
o f A w h i c h h a s o n l y f i n i t e l y many o r b i t s o n A. L e t H b e a subgroup o f (a)
( b ) each
Aut
(r,+)
such t h a t
i s compatible.
(r,A,H) hEH
i s r e g u l a r on
AA.
( c ) H h a s o n l y f i n i t e l y many o r b i t s o n (d)
hEH: h/,EG*.
Then
N = IfEMHu{a) ( r ) I f / A E M G A ( A ) l
0-primitive, a n d t h e DCCL.
b u t n o t 1 - p r i m i t i v e on
nA.
i s zerosymmetric,
r,
has an i d e n t i t y
5 4 PRIMITIVE NEAR-RINGS
120
r9
I f moreover
and A i s a f i n i t e d i m e n s i o n a l v e c t o r
A
tJ
s p a c e o v e r some d i v i s i o n r i n g D a n d i f then
N = If€M,,"~,)(r)lf/,EEndD(r))
r,
b u t n o t 1 - p r i m i ti ve on
"(
NE
hEH:
h/AED
i s also 0-primitive,
no,?i!z 3, , a n d
moreover
i s a ring.
o :A)
4 . 4 1 R E M A R K See a l s o H o l c o m b e ( 4 ) f o r t h e m o r e q e n e r a l c a s e t h a t A i s o n l y a f i n i t e u n i o n o f N-subgroups o f t y p e 2 w i t h zero intersection. 4.42
t h e n i n t h e n o n - r i n g case o f 4.30
GA = { i d )
REMARK I f
we g e t n e a r - r i n g s o f t h e f o r m N = ( f c M o ( r ) l f ( A ) F A ] ( s e e e . g . N o i n 3 . 8 ) . C f . R a m a k o t a i a h - R a o (1),(3),(4). Conversely,
(r,t)
if
i s a f i n i t e group and A a non-
t r i v i a l subgroup then
N:= { f E M o ( r ) l f ( A ) F A }
i s a finite
n e a r - r i n g w i t h i d e n t i t y , z e r o - s y m m e t r i c a n d 0-, b u t n o t 1 - p r i m i t i v e o n r. A i s j u s t t h e s e t o f n o n - g e n e r a t o r s and i s an N-subgroup s u c h t h a t N / ( o : A ) i s a non-ring if
lAI
> 2.
C\
1 - P R I M I T 1 V E N E A R - R I FIGS
Now l e t N b e 1 - p r i m i t i v e o n
r r
r.
C = Go
Then
(by 4 . 1 8 ) ,
i s n o t N-isomorphic t o a p r o p e r subgroup (4.17(bl),
= eouel
R = Col
(by 4.15(d)),
+
t~ L
Ak N , L EOI 3 Y E r : We s t i l l a s s u m e t h a t N E 4.43
THEOREM
Ly =
M(r).
or
r
R =
r
(3.2)
and
(by 3.4(a):.
r . Then Nor f ( 0 ) A R = r . T h e n N o r , Nc = Mc(r) a n d e l = r .
(a) L e t N be 1 - p r i m i t i v e on C a s e 1:
on
If
No
i s a r i n g then
i s 1-primitive
N i s dense i n
Maff(r)
where r i s a v e c t o r space o v e r t h e d i v i s i o n r i n g D : = HomN (F,r). 0
121
4c 1 -primitive near-rings
If
No i s n o t a r i n g t h e n applicable.
(D)
Case 2 : Nor ( 0 1 A R = ( 0 1 . Then m i t i v e on r a n d 4.30 h o l d s . Case 3: Nor = { o l . Then N = N, a s i m p l e g r o u p dp (01.
=
o f 4.30 i s
N = No
Mc(r)
i s 1-priand
r
is
( b ) C o n v e r s e l y , i f a n e a r - r i n g NEM(r) i s such t h a t N o i s 1 - p r i m i t i v e on r w i t h Nce(IO1, M c ( r ) l o r i f N = Mc(r) (r { o } a n d s i m p l e ) then N i s 1 - p r i m i t i v e on r .
+
+
P r o o f . ( a ) I f Nor (01, N o i s 1 - p r i m i t i v e on r by 3 . 1 8 ( a ) . S i n c e e a c h s t r o n g l y monogenic N-group has e i t h e r R = ( 0 1 o r n = r, t h e r e s t f o l l o w s from 1 . 5 0 , 3.9, 3 . 1 5 ( a ) , 4 . 2 7 ( a ) a n d 4 . 3 2 . ( b ) I f No i s 1 - p r i m i t i v e on r and N, = {O) o r 'L N, Mc(r) t h e n e i t h e r R = ( 0 1 o r R = r ( 1 . 5 0 ) , s o N i s 1 - p r i m i t i v e on r by 3 . 1 8 ( b ) and 3 . 1 5 ( a ) . 'L If N Mc(r), r s i m p l e and (01, then c l e a r l y
-
-
N i s 1 - p r i m i t i v e on
r.
+
4.44 R E M A R K 4.43 i s t h e main r e a s o n f o r d e f i n i n g " s t r o n g l y monogenic N-groups r" as i n 3 . l ( b ) a n d n o t by t h e c o n d i t i o n s "monogenic" and ycr: (Ny = R v Ny r)", f o r 4 . 4 3 would n o t be t r u e i n t h i s c a s e : Take 'l = H8, N o : = { f ~ M ~ ( r ) l f ( =P )f ( 6 ) c 1 0 , 2 , 4 , 6 1 A N,: = {fcMc(r)lf(0)c{0,2,4,611. A f ( 4 ) ~ { 0 , 4 ) ) and
'v
T h e n one can show t h a t N : = N o t N c i s a subnear-ring of M ( r ) enjoying the following properties:
Nr
and NO
r
a r e f a i t h f u l , s i m p l e a n d monogenic. Moreover,
ycr: (Ny = R = { 0 , 2 , 4 , 6 ) v Ny = r). B u t (01 f R r, and N o i s n o t 1 - p r i m i t i v e on r ( i t i s n o t e v e n t r u e t h a t f o r a l l y c r Noy i s e i t h e r = { o l , = R o r = r, s i n c e N04 = { 0 , 4 1 ) .
122
8 4 PRIMITIVE NEAR-RINGS
From 4 . 3 0 a n d 4.45
r
= e0u
el
we g e t w i t h a s t r a i q h t f o r w a r d p r o o f
N€TIO
THEOREM L e t t h e n o n - r i n a
r
be 1 - p r i m i t i v e on
but
without %-equivalent generators.
N
Then
i s dense i n t h e n e a r - r i n q
€370
For 1-primitive near-rinqs
{ f c M o ( r ) \ f ( e o ) = {oil.
w i t h D C C we g e t a w h o l e b u n c h
o f i m p o r t a n t r e s u l t s ( c f . Rarnakotaiah ( 3 ) , B e t s c h ( 1 0 ) ) : 4 . 4 6 THEOREM ( B e t s c h (3)).
Let
e n d o w e d w i t h t h e DCCL.
NEWO
be 1 - p r i m i t i v e on 7 and
Then
( a ) T h e r e a r e o n l y f i n i t e l y many % - e q u i v a l e n c e c l a s s e s i f N i s a non-rinq. (b)
3
S E N: ,,N
s. =
N-isomorphic
1 Li,
r)
l e f t i d e a l s and N-groups o f t y p e 1 ( s o
2.50 i s a p p l i c a b l e ! ) ; s =
f i n i t e l y many p a i r w i s e ( t o
Li
i = l
i f N i s a non-rinq then
Ir/%l-i.
( c ) A l l N - g r o u p s o f t y p e 0 a r e N - i s o m o r p h i c a n d o f t y p e 1. (d) N contains a r i g h t i d e n t i t y (not necessarily two-sided). (e)
N is s i m p l e .
( f ) N i s e i t h e r 2 - p r i m i t i v e o n 'I o r t h e r e i s n o N - g r o u p
o f t y p e 2, Proof.
If N i s a ring,
(b)
-
( f) are e i t h e r well-known
o r t r i v i a l . So we w i l l a s s u m e t h a t
No
i s a non-rinq.
( a ) S u p p o s e t h a t t h e r e a r e i n f i n i t e l y many - - e q u i v a l e n c e
v,,vl.y2,...
classes w i t h representatives assume t h a t
i 2 1,
y o ~ e o . Then
hence
(o:yo)=(o:y1)=(o:~y1,y2))3'. d i c t i o n t o t h e DCCL.
( b ) Now l e t
Y
~
~
representatives of
YOEeOl
yl,
(o:yo)
y1yy21...~f31.
Y
.~'yS~
= N
p
.
We may
(o:yi)
for
So b y ( D ) o f 4 . 3 0
.
which i s a contra-
. b e. a c o m p l e t e s y s t e m o f
t h e %-equivalence classes w i t h
...,y S c e 1 .
Then
n
S
( 0 : ~ ~= ) I 0 1
i= 1
,
but
123
4c I-primitive near-rings
minimal l e f t i d e a l s . Now a p p l y 2 . 5 0 ( p )
tf
j E { l ,
...,sl:
to get
N =
'.1 L
Since
r
by 3 . 1 0 .
j = lj * ,-b
L j -N
L.$(o:Y~), J
( c ) Holds by t h e p r o o f o f ( b ) and 3 . 1 1 ( a ) .
N contains a riqht identity
( d ) By ( b ) a n d 3 . 2 7 ( d ) ,
N1 e. sided. (e)
o f 3.8
shows t h a t e i s n o t n e c e s s a r i l y t w o -
3
If I Q N,
L j n I = C O l .
minimal, IE(0:L.) J
. . . ,s l :
jc(1,
=
{O}
But %
(for
L j -N
LjL)I.
Lj
Since
ILjcInL
F), w h e n c e
j
=(O), I = {Ol.
is SO
( f ) By 4 . 7 o r b y ( c ) .
Note t h a t 4.46(a) i s not v a l i d f o r r i n g s : I f 2 s p a c e IR , c o n s i d e r e d a s a n H o m ( r , r ) - m o d u l e , (xELP)
are pairwise inequivalent w.r.t.
n e a r - r i n g which i s p r i m i t i v e on 4.47
=/=
i s the vector all
(1,x)
Hom(r,T)
i s a
a n d h a s t h e DCCL.
N contains a l e f t identity;
COROLLARY N E ' ~ ' ) ~ . D C C N , PI
r
%,
r
I 9 N,
{Ol. T h e n
(a) N !s
1 - p r i m i t i v e c=+
N i s 2 - p r i m i t i v e <->
N i s simple.
( b ) I i s 1 - p r i m i t i v e c->
I i s 2 - p r i m i t i v e <->
I i s maximal.
Proof.
( a ) By 3 . 4 ( c )
and 3.7(c),
2-primitivity coincide.
N i s simple then
If
by 3.4(b)
1 - p r i m i t i v i t y and
I n t h i s case,
I = N
N i s simple.
i s a m i n i m a l i d e a l and
a n d Lemma 3 i n t h e p r o o f o f 3 . 5 4
I : = N; t h e n
A =
r)
(with
N has a f a i t h f u l N-group
Nr
o f t y p e 1, s o N i s 1 - p r i m i t i v e . (b)
follows from (a).
K a a r l i ( 2 ) showed t h a t if N = No i s s i m p l e a n d U i s a m a x i m a l N subgroup of
N w i t h NU
{O} t h e n N i s I - p r i m i t i v e .
K a a r l i ( 4 ) and A d l e r ( 1 ) .
Cf.
also
124
9 4 PRIMITIVE NEAR-RINGS
d ) 2-PRIMITIVE N E A R - R I N G S
A g a i n we assume t h a t i f
N i s 2 - p r i m i t i v e on r then
NCM(r).
1 . ) 2-PRIMITIVE N E A R - R I N G S The s t r u c t u r e o f 2 - p r i m i t i v e n e a r - r i n g s c a n b e d e s c r i b e d as follows.
4.48 THEOREM ( a ) L e t N be 2 - p r i m i t i v e on Case 1:
Nor
I01 A R
r. Then = r. Then
No
i s 2-primitive
N, = M c ( r ) a n d e l = r . No i s a r i n g t h e n N i s d e n s e i n M a f f ( r ) (as i n 4.43); i f No i s a n o n - r i n g t h e n (0) o f 4 . 3 0 i s a p p l i c a b l e ( f o r No).
on
r,
If
Nor 9 { o }
A
R = {ol.
p r i m i t i v e on
r
and 4 . 3 0
Case 2 : Case 3 :
Nor
=
(01.
Then
Then
N
= No
i s 2-
i s applicable.
N = Mc(r)
and
r
is a
c y c l i c group o f prime order. Conversely,
(b)
if
Nce{{O),Mc(r))
No
i s 2 - p r i m i t i v e on
or if
N = Mc(r)
(r
r
o f prime o r d e r ) then N i s 2 - p r i m i t i v e on The p r o o f i s s i m i l a r t o t h e o n e o f 4 . 4 3
4.49
r
a n d if I A
unless order).
I = Mc(r)
r.
and t h e r e f o r e o m i t t e d .
Betsch ( 7 ) ) . I f N i s 2 - p r i m i t i v e { 0 1 , t h e n I i s 2 - p r i m i t i v e o n r, (where r i s n o t a c y c l i c qroup o f prime
T H E O R E M ( c f . F a i n (1) a n d
on
with
a c y c l i c group
N, I
p
125
4d 2-primitive near-rings
NEQ.
( a ) We f i r s t s h o w t h i s t h e o r e m f o r
Proof.
Ir
Ev d e n t l y , Assume t h a t If
3
IA =
i s faithful.
sI r.
A
Iol
then consider
{01.
~ E A : N6
Ir =
But
Therefore
INACIA =
i s f a i t h f u l . Hence
Iol, NA =
A = {o}.
If
I A
+ {ol
N6 =
I
so
(01,
3
then again
NA
If
NA.
6fA:
r
(01, and N A =
{Ol, A
I6
.b
since
sN f , =+
whence
Io1.
16 sN r , s o 16 = Consequently A = r , f o r I6cA. T h e r e f o r e I i s 2 - p r i m i t i v e o n r. ~ ( 1 6 )= (NI)~FIG,
r. Ir
Since
r.
( b ) Now l e t N b e a r b i t r a r y . We may assume t h a t
N
No,
=/=
I s excluded.
so case 2 o f 4.48
I f N f a l l s i n t o c a s e 1, No i s 2 - p r i m i t i v e o n r . BY 2.18 , I , = ~n ri, 2 N 0 ' I f I , p (0) t h e n I, i s 2 - p r i m i t i v e o n r , hence
I i s 2 - p r i m i t i v e on r by 3.18(b). I f I, = {Ol t h e n I e N c = Mc(r). S i n c e el = r , 10 AN r . 10 = { o l i m p l i e s t h a t f o r a l l y = n o c r and f o r a l l i E I i y = ino = 0 , so I = I o ) .
Hence
10
T a k e any
3
+ Io)
and so
mceMc(T);
is1 : io =
u.
10 =
m o =:p. C
r. Because o f
10 = ,'I
= i o = p = mco = mcy, h e n c e i = mc % I = Mc(r). If 3 P E P : r Zp, I is 2 - p r i m i t i v e o n r; i f not, I i s not 2-primitive. Now
yfr:
iy
-
a n d we g e t
If N i s i n c a s e 3, 4.50
REMARK
I i s t r i v i a l l y 2 - p r i m i t i v e on
4 . 4 9 c a n n o t b e t r a n s f e r r e d t o 0- o r 1 - p r i m i t i v i t y ,
n o t even f o r f i n i t e ,
abelian, zerosymmetric near-rings.
I t i s e a s y t o show t h a t i f N i s e . g . O - p r i m i t i v e o n I 9 N t h e n I r i s f a i t h f u l and monogenic. B u t n o t necessarily simple: T a k e r: = z 8 , A : = IO,2,4,61 a n d E: = I 0 , 4 l . N:
r.
= {ffMo(r)lf(A)rA
A f(5)=f(l)
h
f(7)=f(3)),
r
and
I:= (0:A).
9 4 PRIMITIVE NEAR-RINGS
126
N i s 0 - p r i m i t i v e on
r,
but
E d1
r.
Moreover,
s t r i c t l y monogenic and I has a r i q h t i d e n t i t y . e v e n b e 0 - p r i m i t i v e o n some o t h e r g r o u p Assuming t h a t ,
take
and ( 0 : 3 ) 1
I"
Ir
is.
I cannot
=:IyA:
Ir
i n
and p u t
L : = ( 0 : ~ ; ) ~Then . L i s a maximal l e f t i d e a l o f I ( 3 . 4 ( f ) ) L (0:1)1 , L ( 0 : 3 ) 1 ( s i n c e ( 0 : 1 ) 1 and (0:3)1 c a n n o t be m a x i m a l ) . T h e r e f o r e ( o : l ) I + L = ( f l : l ) I+ L = I . b u t ( 0 : l ) n (0:3) = E ~ I c - L , s o I w o u l d h a v e t o b e a r i n g
and
by 3.4(i),
a contradiction.
So p o i n t e d o u t , N N =
A s Y.S.
I
(0:l) n(0:3).
S e e m i n g l y t h e r e i s no "smal l e r " c o u n t e r e x a m p l e t h a n t h a t above w i t h 4096 e l e m e n t s . See a l s o 5 . 1 9 ( a ) . By t h e way, case
o n e c a n u s e Z o r n ' s lemma t o show t h a t i n a n y
I 9 N (I9 IO), N w - p r i m i t i v e )
h a s some I - g r o u p s
o f t y p e v. 4 . 5 1 COROLLARY ( F a i n ( 1 ) ) . L e t P b e a 2 - p r i m i t i v e i d e a l o f NET,. L e t I b e a n o t h e r i d e a l o f N c o n t a i n i n g P . Then P i s a 2-primitive
Proof. I
9 P,
A NIP,
ideal of I . and
NIP
'Ip p I01
and
i s 2-primitive.
Since 'Ip i s 2 - p r i m i t i v e . Hence
P i s 2-primitive i n I.
2 . ) 2 - P R I M I T I V E NEAR-RINGS I.IITH IDENTITY
I n t h i s case, Also,
if
N = No
(if
=
then
%
=
%
%
NE~,)
or
el
=
r
( i fN $ % ) .
(by 4.21).
Recall that a 1-primitive near-ring cfl0 with a l e f t identity i s a l r e a d y a 2 - p r i m i t i v e one w i t h i d e n t i t y ( 4 . 6 ( b ) ) . We a r e now i n a p o s i t i o n t o g e t a " r e a l " a n d f u n d a m e n t a l d e n s i t y theorem.
4d 2-primitive near-rings
N f N,
N
(case 1
No a non-ring
No
(case 2 o f
o f 4.48)
No a r i n q
=
127
4.48)
-
Ma f f ( r )
HomD ( r ,r 1
Dr a v e c t o r s p a c e $. fol
l'fc (r)
M C ( r ) = MC(r)
Go f i x e d - p o i n t -
0
0
r
f r e e on
(b) Conversely,
e v e r y n e a r - r i n g w h i c h i s dense i n Maff(r) o r HomD(r,r) (where r i s a non-zero v e c t o r space o v e r some d i v i s i o n r i n g D ) o r d e n s e i n (r) o r
MC 0
on Proof.
(r)
r,
fixed-point-free
(Go
where Co = G
w
0
2 - p r i m i t i v e on note that
1
= MC
r
(r)
4.28(c)
and has an i d e n t i t y .
No
i s therefore I f No i s a i s not a rinq,
on
r,
= MGoV{b}
(r)
since
Y I sN r .
then
My;
0
(r)
o(r)
= M
=
Go
(D)
o f 4.30,
and 4 . 2 7 ( a ) .
Mo(r),
=/=
If
and t h e r e s u l t f o l l o w s f r o m
equal t o )
If N
No
0
Go = { i d }
in
and has an i d e n t i t y
cannot occur.
i s fixed-point-free
Go
+ {id}
Go
0
r
CvcrJs(Y) =
gE:Go:
If
If
{61.
t h e s t a t e m e n t is c l e a r .
ring,
i s 2-primitive
on 7 )
( a ) I f N i s 2 - p r i m i t i v e on
t h e n case 3 i n 4.48
mc 0
No,
then (D)
of 4.30
implies that
N
which i s t r i v i a l l y dense i n ( s i n c e
MC
0
(r)
= MCal(r)
=
Mo(r).
apply again 4.27(a).
0
i s dense
8 4 PRIMITIVE NEAR-RINGS
128
( b ) Assume now t h a t N i s d e n s e i n H o m D ( r , r ) , where i s a n o n - z e r o v e c t o r s p a c e o v e r some s k e w - f i e l d D.
r
N i s a dense s u b r i n g and t h e r e f o r e a p r i m i t i v e
Then
r i n g o n r . F r o m t h i s we d e d u c e : I f N i s dense i n M a f f ( T ) then
i s dense i n
No
s o r h a s no n o n - t r i v i a l N o - s u b g r o u p s , HomD(r,r), a n d N i s 2 - p r i m i t i v e o n r. I f N i s d e n s e i n M O(r) GO
each dense s u b n e a r - r i n g of t h a t i s t r i v i a l l y 2 - p r i m i t i v e o n r. I f G o {id} t h e n M o(r) = FIG ( r ) a n d 4.28(c)
0
GO
Nr c a n n o t c o n t a i n n o n - t r i v i a l
shows t h a t
N-subgroups. Finally i f
in
(Tic
0
N i s dense i n
(r))o
a non-trivial
MC (r)
then
0
As above,
= MC ( I - ) .
r
No
i s dense
cannot c o n t a i n
0
No-subgroup (or one can use 3 . 1 8 ( b ) ) .
4.53 REMARKS (a) I t i s n o t t r u e t h a t each 2 - p r i m i t i v e n e a r - r i n q w i t h identity, N take
MC(r): and
N: =
and
No
a non-ring,
i s dense i n
=/=
No
r
f i n i t e w i t h I i d l f G s A u t ( r ) , G f i x e d - p o i n t free Nr h a s R = r, s o
F i G ( r ) . Then
CN(r) = { i d ) ( 4 . 1 3 ( c ) ) a n d t h e r e f o r e MC(r) = M(r). But N M ( r ) , s o N c a n n o t be d e n s e i n M ( r ) b y
+
4.29.
This i s a l a t e b u t convincing reason f o r
introducing t h i s crazy s w i t c h e s down t o
No
N, ( r ) ,
w h e r e one f i r s t
0
(by forming
Co = End
NO
(r)
t h e n back up b y a d d i n g a l l o f t h e c o n s t a n t s :
and
MC(r)
would be t o o b i g i n general. ( b ) 4 . 5 2 ( a ) does n e i t h e r h o l d f o r 0 - p r i m i t i v e n e a r - r i n g s with i d e n t i t y nor f o r 2-primitive near-rings without i d e n t i t y ( n o t even f o r
N =
No
and N f i n i t e ) :
129
4d 2-primitive near-rings
r:
n4
N : = {fcMo(T)\f(A)!!A) i s 0 - p r i m i t i v e on r w i t h i d e n t i t y , b u t n o t dense i n M C ( r ) = Mo(r) ( 4 . 2 9 ! ) . M: = { f E M o ( r ) l f ( 3 ) = 0 1 i s If
=
and
A:
= (0,2),
0
2 - p r i m i t i v e on
MC (r)
in
r,
w i t h o u t i d e n t i t y and a g a i n n o t dense
Mo(r).
=
0
(c) A l l 2-primitive near-rinas with i d e n t i t y on w h e r e No ( d ) 4.32,
i s a non-ring,
4.43
and 4.48
Z4,
w i l l b e c l a s s i f i e d i n 4.63.
reduce t h e theorv o f p r i m i t i v e
near-rings t o those o f p r i m i t i v e zero-symmetri c nearr i n g s . We w i l l t h e r e f o r e m a i n l y d e a l w i t h t h o s e o n e s
i n t h e sequel. (e) Recall (4.27(a))
that
Kc
general. H e r e i s some e x a m p l e :
= -x)
r
A!
=
i s "only a set" i n
(f) 0
G = {id,-idl
i s a fixed-point-free
IR.
= {o,id,-id}.
C:
XER:
f(-x)
(with
-id(x):
=
autsmorphism group on
Vc(IR)
= {fEM(IR)If(O)
= 0 A
-f(x)).
n If m c ( R ) = : N , t a k e nl: = sin+-pN and n2: = n C o n s i d e r n : = nlon2EM(IR ) . = id+pN. n no = s i n o ( i d t r ) - s i n ( % ) i s n o t an odd f u n c t i o n , t h u s n o t b e l o n g i n g t o MC(IR), w h e n c e ngN a n d N i s no n e a r - r i ng.
4.54
COROLLARY I f N i s 2 - p r i m i t i v e o n
r
with
AutN 0
(r)
= {id)
t h e n N i s dense i n e i t h e r one of t h e f o l l o w i n g n e a r - r i n o s ( n o t a t i o n as i n 4.52): H o m D ( r , r ) , Maff(r), Mo(r) o r
M(r) 4.55
(cf.
4.65).
THEOREM ( R a m a k o t a i a h non-ring on
r
classes w.r.t.
(2)).L e t
NET),
be a 2 - p r i m i t i v e
w i t h an i d e n t i t y . Then any two e q u i v a l e n c e
2
(except t h e zero class) are equipotent.
$4 PRIMITIVE NEAR-RINGS
130
T*/%
Proof. L e t E be i n
C o n s i d e r t h e map
and
a f i x e d element o f E.
E
f: G
+
E
9
+
q(E)
(with
Since G i s fixed-point-free
(4.52),
aqain).
f i s injective.
so f i s a b i j e c t i o n .
f i s surjective,
By d e f i n i t i o n ,
G=AutN(r)
3 . ) 2 - P R I M I T I V E Z E R O - S Y M M E T R I C N E A R - R I N G S WITH I D E N T I T Y A N D A M I N I M A L LEFT I D E A L . 4 . 5 6 THEOREM ( B e t s c h ( 7 ) , c f . D e s k i n s ( 2 ) ) . w i t h i d e n t i t y which i s 2 - p r i m i t i v e on
r
L e t N = N , b e a nr. and has a m i n i m a l
l e f t i d e a l L . Then %
(a) L = N r' ( b ) 3 e2 = eeL*:
L = Ne = L e
and
morphic t o Proof.
(a) Since
and
+ {ol,
Lr
3
W i t h y as a b o v e , and
m i ni m a l ) . k-%e
Pie
E
i s antiiso-
(eNe;). YEr:
+ {ol,
Ly
so
Ly =
r
Now we c a n a p p l y 3 . 1 0 .
yeel.
= ey
3
(C,,,(T),a)
Hence
(o:y)nL
= Le = L.
eEL":
e y = y.
=
and
e2 = e
{O} a n d L e
By ( a ) ,
Therefore
I01
(since L i s
Le
{Ol.
e2-eEL n ( o : y ) =
SN L,
Since
Le = L .
By L e C N e s L ,
C N ( r ) = CN(L) = CN(Ne)
( i t can
be e a s i l y v e r i f i e d t h a t N - i s o m o r p h i c N-groups have i s o m o r p h i c centralizer-semigroups). For
neN,
consider
d e f i n e d and
t n : Ne xe
+
+
Ne xene
.
i s well-
tn
E C ~ ( N ~ ) .
C o n s i d e r n e x t t h e map
h:
eNe ene
+ +
CN(Ne). tn
If
e n e = erne
so h i s w e l l - d e f i n e d . C l e a r l y , t h e n t n = t,, h i s an a n t i h o r n o m o r o h i s m . I f h ( e n e ) = h ( e m e )
t n = t,
and
Specializing
then
tj X E N : x e n e = t n ( x e ) = t m ( x e ) = xeme. x = :e
we g e t
e n e = eme
and h i s
shown t o b e i n j e c t i v e . Finally,
b
c€CN(IJe)
3
nEN:
c ( e ) = ne.
ene = e c ( e ) = c ( e 2 ) = c ( e ) = ne aet
c ( x e ) = x c ( e ) = xene
tn(xe),
Therefore
a n d f o r a l l X E N we
so c = tn.
4d 2-primitive near-rings
131
and h i s s u r j e c t i v e , hence an a n t i i s o m o r p h i s m . COROLLARY ( B e t s c h ( 7 ) ) .
4.57
N ~ r t ) h a s an i d e n t i t y a n d
If
a minimal l e f t i d e a l L t h e n a l l f a i t h f u l N-groups o f type 2 ( i f those e x i s t ) a r e N-isomorphic
(r,
determines t h e p a i r
I f e i s as i n 4.56(b),
t h e group
CN(r))
u n i q u e l y "up t o i s o m o r p h i s m " .
i s a group w i t h z e r o and L = Ne a s a f i k e d - p o i n t -
(eNe,.) a c t s on
(eNe\{O),*)
( t o L ) and N
f r e e a u t o m o r p h i s m g r o u p (by r i g h t m u l t i p l i c a t i o n ) . Hence e " b r i n g s b a c k " some i n f o r m a t i o n o n 4.58
r
out of NSM(T). C f .
REMARK F o r m o r e i n f o r m a t i o n o n t h e s e t o p i c s ( a p a r t i a l c o n v e r s e o f 4 . 5 6 , t h e u n i q u e n e s s o f (r, CN(r)), s e e B e t s c h ( 6 ) a n d 5 7 a ) , i n p a r t i c u l a r 7.5.
4.)
4.59
COROLLARY L e t 2.50
( f o r "),
(2.50(a)
be a 2 - p r i m i t i v e n e a r - r i n g w i t h
NEW,
5
i s applicable,
hence a l s o
a n d G h a s f i n i t e l y many o r b i t s o n
and 4.21),
i s d i s c r e t e "on
r,
w h i c h i s t h e same a s t o s a y t h a t
MG(r)
Z
(4.29).
7a) f o r the information t h a t i f a f i x e d - p o i n t - f r e e
automorphism group H o f
r
h a s f i n i t e l y many o r b i t s o n
MC(r) h a s t h e DCCL. See a l s o K a a r l i 4.60
etc.)
2 - P R I M I T I V E P!EAR-RINGS WITH I D E N T I T Y A N D MINIMUM CONDITION
DCCL a n d i d e n t i t y . T h e n 4 . 4 6
See
9.227.
r
then
( 2 ) and Oswald ( 1 0 ) .
THEOREM ( B e t s c h ( 7 ) ) . L e t N b e 2 - p r i m i t i v e o n r w i t h D C C f o r t h e l e f t i d e a l s o f No and w i t h a n i d e n t i t y . Then N i s equal t o one o f t h e f o l l o w i n g n e a r - r i n g s ( n o t a t i o n as i n 4.52):
NpNo
N - N o
No a r i n g
Maff(r) H o m D ( r , r )
No a n o n - r i n g
Mc ( r ) 0
MC(r)
dimDr f i n i t e Go h a s f i n i t e l y many
o r b i t s on
r
132
$ 4 PRIMITIVE NEAR-RINGS
P r o o f . f o l l o w s from 4 . 5 2 and 4.59.
Note t h a t
Rc (I') 0
i s a n e a r - r i n g i n t h i s case ( f o r i t equals N ) . 4 . 6 1 COROLLARY I f N h a s a n i d e n t i t y , i s 2 - p r i m i t i v e o n r a n d i f t h e n o n - r i n g No h a s t h e DCCL a n d A u t N ( r ) = ( i d ) 0
N = M(r) ( i f N No) or o t h e r w i s e N = Mo(r). I n b o t h c a s e s , r ( a n d t h e r e f o r e N, t o o ) i s f i n i t e . So t h e D C C i m p l i e s f i n i t e n e s s ! then e j t h e r
4.62
R E M A R K These r e s u l t s i l l u s t r a t e some r e m a r k s i n t h e
preface: w h i l e the "elements o f r i n q theory" a r e r i n g s
o f l i n e a r mappings on r , t h o s e ones f o r n e a r - r i n q t h e o r y a r e n e a r - r i n g s o f a r b i t r a r y mappings ( p e r h a p s w i t h some r e s t r i c t i o n s ) o n
r.
4 . 6 3 THEOREM ( K a a r l i ( 4 ) ) I f
-
then
I a-S z- N
and i f S / I
i s 2-primitive
I d- N .
Proof. Since I i s a 2 - p r i m i t i v e l e f t i d e a l o f S,
I = (L:S)s
h o l d s f o r some 2 - m o d u l a r l e f t i d e a l L o f S b y 4 . 3 . By 3 . 3 4 ,
L 2NS.
Consequently,
Hence
I
=
(L:SINn S
and ( L : S l N 2 N .
I i s an i d e a l o f N .
See a l s o K a a r l i ( 2 ) a n d R a m a k o t a i a h ( 2 ) .
I n t h e l a t t e r paper i t
i s shown t h a t i f N€Hl i s f i n i t e a n d 2 - p r i m i t i v e o n
r,
if N i s a
lrl-1 i s a p r i m e t h e n e i t h e r N=M(T) o r N=Mo(T) n o n - r i n g and i f or ~ N ~ = ~ i r f~ zr ; i s a b e l i a n , N z N rOr h o l d s i n t h e l a s t c a s e ( t h i s r e s u l t can be deduced f r o m 4.55
and 4.61).
133
4d 2-primitive near-rings
5.)
AN APPLICATION TO INTERPOLATION T H E O R Y
4.64
reg
DEFINITION I f
and
NEM(r),
N i s said t o f u l f i l l
the f i n i t e interoolation property I f
w
SE:nU
3
ncN
v
w
Y1#...*YSEr,
Yi
sl:
n(yi)
i c { l #...,
t
Yj
9
for
\I
j
61’..-’6sEr
= 6i.
T h e r e i s an o b t r u s i v e s i m i l a r i t y t o t h e d e n s i t y c o n c e p t s .
In
fact: 4.65
Let
THEOREM
N I M(r)
with
N
and
No
No
not a ring.
Then t h e f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t :
r*,
( a ) No i s 2 - f o l d t r a n s i t i v e o n
r
( b ) N i s 2 - p r i m i t i v e on
Go = { i d ) .
with
( c ) N f u l f i l l s the f i n i t e i n t e r p o l a t i o n property.
Proof.
( a ) ->
(b):
Nr i s t r i v i a l l y f a i t h f u l , 2 - f o l d
t r a n s i t i v i t y implies 1-fold i n turn that Some
6cA”
No6 =
r
and
Nr
{ol.
If
W
ycr
Then A =
r.
t r a n s i t i v i t y and t h i s
I o l 9 A sN r , 0
2
n,ENo:
+
take
no6 = y .
9
So
If Goy 5: G o 6 but y 6 and ( s a y ) 6 o, take 0 . Then noEN 0 w i t h n o y = o A no6 (0:6) , s o y+6 i n (0:Y) NO NO Nor * h e n c e (4.20(c)), a contradiction, Therefore Goy 9 G o 6 some
+
Go = { i d } .
(b)
(c):
b y 4.54
( c ) ->
(a):
trivial.
K a i s e r ( l ) , Lausch ( 5 ) , Ramakotaiah ( 3 ) .
Cf.
and 4.28(d).
P l i t z (12),(13),
P i l z ( 2 5 ) and
134
5 4 PRIMITIVE NEAR-RINGS
4 . 6 6 REMARKS
( a ) So i f a n e a r - r i n g N o f m a p p i n g s o n r i n t e r p o l a t e s a t o and 2 o t h e r p l a c e s then N i n t e r p o l a t e s a l r e a d y on an a r b i t r a r y ( f i n i t e ) n u m b e r o f p l a c e s . Compare t h i s w i t h the corresponding "1 i n e a r " r e s u l t i n r i n g theory ( ( N . J a c o b s o n ) , C o r o l l a r y t o theorem 1 on p . 3 2 ) . This i s a " p u r e l y i n t e r p o l a t i o n - t h e o r e t i c " r e s u l t . ( b ) I t c a n be s h o w n t h a t i f N f u l f i l l s t h e f i n i t e i n t e r p o l a t i o n p r o p e r t y a n d 1i-l > 3 t h e n N o i s a non-
ring. 4 . 6 7 C O R O L L A R Y T a k e r = (IR,+). Then a n y o n e o f t h e f o l l o w i n g n e a r - r i n g s and a l l n e a r - r i n g s c o n t a i n i n q o n e o f t h e m h a v e t h e p r o p e r t i e s t h a t Pi i s 2 - p r i m i t i v e on r w i t h N No, Go = { i d ) and No n o t a r i n g :
+
[XI,N 2 :
t h e n e a r - r i n g o f a l l s t e p f u n c t i o n s on IR the subnear-ring of M(1R) g e n e r a t e d by t h e t r i g o n o metric polynomials.
N1: N3:
= IR
F o r a l l o f them f u l f i l l t h e f i n i t e i n t e r p o l a t i o n p r o p e r t y w h i c h q u a l i f i e s them f o r 4 . 6 5 . The a u t h o r h o p e s t h a t n e a r - r i n g s o f i n t e r p o l a t i n g f u n c t i o n s become i n t e r e s t i n g f o r s p p r o x i m a t i o n t h e o r y ( b e c a u s e t h e s e f u n c t i o n s c a n be i t e r a t e d w . r . t . 0 ) . After a l l t h a t complicated s t u f f the r e a d e r will p o s s i b l y a g r e e w i t h t h e a u t h o r t h a t t h e p r i m i t i v e n e a r - r i n q s have s u c c e s s f u l l y r e v e n g e d t h e i r d i s c r i m i n a t i n g name.
,
$ 4 PRIMITIVE NEAR-RINGS
This paranraph presents a discussion o f t h e " b u i l d i n q stones, near-rinqs
a r e made o f " ,
Similar t o r i n q theory,
the so-called t h e "atoms"
"primitive near-rinqs".
a r e n o t t h e simple near-
r i n g s as one m i o h t e x p e c t a t a f i r s t g l a n c e . an i m p o r t a n t c o n n e c t i o n (4.47).
g i v e n a n e a r - r i n q N,
f r u i t s (= N-groups)
r e c o q n i z e them b y
we l o o k a t a l l o f i t s
and ask, w h e t h e r t h e r e a r e f a i t h f u l and
' ' e n o u g h s i m p l e " o n e s amonq t h e m . N " p r i m i t i v e on t h i s N-group". p r e c i s e we f i x
however,
The i d e a t o c o n s i d e r p r i m i t i v e
n e a r - r i n g s comes f r o m t h e b i b l e ( " Y o u w i l l their fruits"):
There i s ,
If t h i s i s t h e c a s e , we c a l l
S i n c e "enouqh s i m p l e "
i s not
i t s menninq i n w a n t i n g N-qroups o f t y p e v.
The r e s u l t i n q c o n c e p t i s t h a t o f " v - p r i m i t i v i t y " . We g e t t h e h i e r a r c h y 2 - p r f r n i t i v i t y < T > l - p r i m i t i v i t y < Z ' mitivity,
discuss conditions,
O-pri-
w h i c h f o r c e some o f t h e s e c o n c e p t s
t o c o i n c i d e a n d make a l o t o f w o r k t o w a r d s a d e n s i t y t h e o r e m which i s c o m p a r a t l e t o t h e c e l e b r a t e d one i n r i n o t h e o r y due
t o N . J a c o b s o n . We r e a l l y q e t o n e f o r 2 - p r i m i t i v e n e a r - r i n q s w i t h i d e n t i t y (4.52).
A d d i n q a c h a i n c o n d i t i o n , we a r r i v e a t
a Wedderburn-Artin-like
s t r u c t u r e theorem (4.60).
Before that,
we g e t " b e t t e r a n d b e t t e r " d e n s i t y - l i k e s t r u c t u r e t h e o r e m s f o r 0-,
1- a n d 2 - p r i m i t i v e n e a r - r i n g s .
theorems on v - p r i m i t i v e symmetric v - p r i m i t i v e
I t comes o u t t h a t m a n y
n e a r - r i n q s c a n be d e r i v e d f r o m z e r o -
n e a r - r i n q s w h e r e t h e y a r e much e a s i e r
t o o b t a i n s i n c e t h e s e ones behave more l i k e r i n q s . many p r o o f s c o n c e r n i n g e v e n z e r o - s y m m e t r i c
However,
near-rinqs
differ
t o t a l l y f r o m t h e comparable ones i n r i n g t h e o r y . Anyhow,
t h e " b u i l d i n g stones" mentioned above ( 2 - p r i m i t i v e
n e a r - r i n g s w i t h i d e n t i t y ) a r e shown t o b e d e n s e i n or
Maff(r) ( i f
MGo"k51 ( r ) t M c ( T ) fixed-point-free
No
i s a ring) or i n
( i f No
MGo,{al
i s a non-ring),
automorphism group
where
(r).
AutN 0
HomD(r,r)
(r)
or
Go
i s the
I n particular,
4a General
103
i f Go = ( i d ) , t h e l a t t e r t w o o n e s a r e Mo(r) a n d M ( r ) . F i n a l l y , t h e d e n s i t y p r o p e r t y i s s e e n t o be a k i n d o f a n i n t e r p o l a t i o n p r o p e r t y and a " p u r e l y i n t e r p o l a t i o n - t h e o r e t i c " r e s u l t w i l l be o b t a i n e d . R e c a l l a g a i n ( p . 1 ) t h a t r * = r \ { o ) , a n d so o n .
a 1
G E N E R A L
1 . ) DEFINITIONS A N D E L E M E N T A R Y RESULTS 4.1
CONVENTION I n a l l w h a t f o l l o w s , w w i l l be a n y n u m b e r unless otherwise specified.
E I O , ,Zl ~ 4.2
DEFINITION
( a ) 1.4 i s c a l l e d u - p r i m i t i v e on
Nr:
<->
Nr
is f a i t h f u l
o f type w.
y:
( b ) ri i s v - p r i m i t i v e : <=> 3 N r E N N i s w - p r i m i t i v e on ( c ) I A N i s c a l l e d a u - p r i m i t i v e i d e a l o f N : <-> N/I w - p r i mi t i ve . 4.3
Nr. is
T h e n the following
PROPOSITION L e t ' I be an i d e a l o f N . conditions are equivalent: ( a ) I i s w-primitive. (b)
3 NrENq:
I =
(o:r)
( c ) 3 L s$, N : I = ( L : N ) Proof.
( a ) ->
Nr is
A A
o f type w.
L i s v-modular.
( b ) : I i s w - p r i m i t i v e -> N/I i s u - p r i m i t i v e N / I r -> Nr ( a s i n 3 . 1 4 ( b ) ) i s o f t y p e w a n d
o n some I = (o:r).
( b ) -> ( c ) : L e t r be = Ny p I o l . ( 0 : ~ )= : L . Then 'L L i s m o d u l a r . By 3 . 4 ( e ) , N / L r , so L i s w-modular. Finally, I = ( o : r ) = (o:N/L) = (L:N). nN
( c ) => ( a ) : T a k e N / L = : r . T h e n ( a s above) I = (L:N) = ( o : r ) .
Nr
i s o f type w and
8 4 PRIMITIVE NEAR-RINGS
104
4.4
COROLLARY
The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t :
(a) N i s w-primitive. (b) (c)
4.5
I01
3
i s a w-primitive ideal.
L 3% N :
t w-modular A (L:N)
IOI.
=
REMARKS ( a ) Observe t h a t ( c ) i n 4.3 and 4 . 4 q i v e " i n t r i n s i c " characterizations o f primitivity
-
t h a t w i l l be
f o r i t e n a b l e s one t o r e c o g n i z e
extremely helpful,
p r i m i t i v i t y " w i t h i n N". (b) I f
N i s v - p r i m i t i v e on r t h e n IA N
if
i s a w-primitive
(c) 2-primitivity
Nv
( d ) The n e a r - r i n g s (e)
ideal then
r
Io?
1 4
N.
and
i m p l i e s 1 - p r i m i t i v i t y and t h i s i n t u r n
implies 0-primitivity n e a r - r i nqs.
N $. f 0 1 ,
( a l w a y s o n t h e same g r o u p ) .
o f 3 .8 a r e e x a m p l e s o f w - p r i m i t i v e
(on Z4).
If N i s v-primitive
on
r
then
M(r)
N 4
(1.48).
( f ) See 95 o f B e t s c h ( 3 ) f o r a d i s c u s s i o n o f t h e s p a c e s o f w-primitive 4.6
PROPOSITION L e t i d e n t i t y e.
( v = 1,2)
ideals
of
NEWo.
N contain either a l e f t or a riqht
Then
(a) Every u - p r i m i t i v e i d e a l I o f N i s modular.
(b) I f e i s a l e f t i d e n t i t y o f N t h e n N i s 1 - p r i m i t i v e iff N i s 2-primitive
(and i n t h i s case e i s a two-
sided identity). Proof.
( a ) I f e i s a l e f t i d e n t i t y i n N t h e n (because
N/I
i s w - p r i m i t i v e o n some
of
N/I
by 3.4(c).
So
N/Ir) e + I
\f nEN: e n
f
i s an i d e n t i t y
n e E n {mod I ) .
If e i s a r i g h t i d e n t i t y , t h e a s s e r t i o n i s t r i v i a l . (b) L e t N be I - p r i m i t i v e on 3.4(c),
Nr
e i s a two-sided
i s unitary.
Nr.
By
i d e n t i t y f o r N.
Flow a p p l y 3 . 7 ( c )
By 3 . 4 ( b ) ,
and 3 . 1 9 ( a ) .
105
4a General
4.7
PROPOSITION L e t FI be s i m p l e a n d N i s v - p r i m i t i v e on r. P r o o f . (o:r) 4 N , so the c o n t r a d i c t i o n
4.8
Nr
(o:r) = to1 Nr = { o l ) .
b e o f t y p e w . Then
(for
(o:r) = N
gives
PROPOSITION ( B e t s c h ( 3 ) ) . L e t t h e r i n g N be w - p r i m i t i v e on r . T h e n N i s a p r i m i t i v e r i n g on t h e N-module r ( ( N . J a c o b s o n ) , p. 4 ) .
Proof, I f
r
= Ny
abelian. If nEN:
then
% r -N N/(o:y)
nl+(o:$),
(r,+)
is
n 2 + ( o : y ) ~ N / ( o : y ) then
n(nl+(o:y)+n2+(o:y))
= n ( n , t ( o : y ) ) + n ( n 2 + ( o: y ) )
v
and
.
= nnlt(o:y)+nn2t(o:y)
=
Hence tf y l , y 2 ~ r nEN: n(y1ty2) nyltny2, and N r i s a ( r i n g - ) module. Each N-submodule o f N r i s a n i d e a l , s o = l o ) o r = r . F i n a l l y , Nr p { o ) by a s s u m p t i o n , s o Nr i s i r r e d u c i b l e a n d N i s p r i m i t i v e on r . 4.9
C O R O L L A R Y (Ramakotaiah ( 1 ) ) . I f N i s commutative and u - p r i m i t i v e then N i s a f i e l d .
LEX~(N):
(L:N) = {OI. L 9 N , s i n c e P r o o f , BY 4 . 4 ( ~ ) , 3 N i s c o m m u t a t i v e . By 3 . 2 5 , (L:N) is the greatest i d e a l i n L, s o L = (01 a n d N c o n t a i n s a r i g h t i d e n t i t y . By 1 . 1 0 7 ( c ) , N i s a r i n g , h e n c e a p r i m i t i v e r i n g by 4 . 8 a n d by ( N . J a c o b s o n ) , p . 7 a f i e l d . In ( 3 ) , R a m a k o t a i a h s h o w s t h a t i f I q N = N o a n d I €g0(t\1) t h e n I is a 0 - p r i m i t i v e ideal. Near-rinas N with a f a i t h f u l , s i m p l e , n o n t r i v i a l N-group a r e c a l l e d ? - p r i m i t i v e a n d a r e s t u d i e d in H a r t n e y ( 4 ) , M e l d r u m ( 7 ) , ( 1 3 ) . S e e a l s o B e i d l e m a n (7),(8),(9). H o l c o m b e blalker ( 1 ) s t u d y 3 - p r i m i t i v e n e a r - r i n q s PI, w h i c h m e a n s t h a t N h a s a f a i t h f u l N-group o f t y p e 3 ( s e e t h e l a s t l i n e s o f p. 80).
8 4 PRIMITIVE NEAR-RINGS
106
2 . ) T H E CENTRALIZER
4 . 1 0 DEFINITION E n d N ( r ) = H m N ( r , r ) =: C N ( r ) = : C i s c a l l e d t h e c e n t r a l i z e r o f Nr ( c f . K a a r l i ( 2 1 , R a m a k o t a i a h ( 3 ) ) .
A u t N ( r j =: G o .
A u t N ( r ) =: G N ( r ) =: G;
0
Go:
if
=
~ E C ;
otherwise 4 . 1 1 REMARKS ( a ) (C,.)
C Q = {o} ( b ) ~ E <=> (c) If 4 . 1 2 NOTATION
(ci,o)
.
N 4 M C N ( r ) (l-1 5
is f a i t h f u l then
NT
If
fn:
nEN,
l i k e w i s e .:G
a n d (Go,") are q r o u p s , ( " g r o u p s w i t h z e r o " ) a r e rnonoids.
i s a monoid, a n d (Go,,")
(GO,")
;
r
-+
'I
Y
-c
ny
;
FN(r):=
M&9.
ffnlnENl =: F .
4 . 1 3 PROPOSITION ( M l i t z ( 3 ) ) .
(a) If
= ImcM(r)I
(b)
tl
hECN(r):
(c) If Proof.
tl
CN(r)= fom) = : MF(r).
Nr i s m o n o g e n i c t h e n
Q =
r
(a) If YET:
WfEF: mof = h/n = id.
then
CN(r) = { i d } .
hECN(r)
and
fnEFN(r)
then
(hofn)(y) = h(ny) = nh(y) = ( f n o h ) ( y ) ;
so
hEMF(r).
sMF(r). I f y 1 , y 2 & r = Ny a n d y 1 = n l y A y 2 = n 2 y . Then
C o n v e r s e l y , l e t f be
n&N
then
f(nv1) =
3
nl,n2EN:
(fof,)(vl)
=
(fnof)(vl)
= nf(Y1)
and
107
4a General
Hence
feCN(r).
(b)V
heCN(r)
noc:R: h ( n o ) = n h ( o ) = n o .
( c ) f o l l o w s from ( b ) . 1.14 N O T A T I O N
eo:
=
e o ( N r ) := { y c r
el:
=
el(Nr):
Ny = NO =
Rl.
ri.
= C Y E ~ NY =
1 . 1 5 REMARKS ( B e t s c h ( 6 ) ) .
(a)
oEeo,
e l $. 0 ( c ) eon e l (b)
P 0.
so
eo
<->
Nr i s m o n o q e n i c . n $. r .
= 0 <->
r
( d ) Nr i s s t r o n q l y m o n o g e n i c -> ( e ) Nr
eo
i s u n i t a r y ->
and
=> y = l y c R
( f ) G ( B o ) = Bo g r o u p s on
=
A G(B1)
eo
and
(for
= R
w =
= Bow
el,
y e e 0 ->
Ny
R ==>
nocR => Nu = NnocNo = R =I>
so G i n d u c e s p e r m u t a t i o n
el, (if
WE^^)
+ 0)
el
on
el.
The n e x t p r o p o s i t i o n i s a " S c h u r - t y p e lemma". 4 . 1 6 P R O P O S I T I O N ( B e t s c h (S), M l i t z ( 3 ) ) . ( a ) Nr i s s i m p l e (hEC A 2
(b)
n
A
yEB1:
= { o } ->
h(y)EOl)
Nr i s N - s i m p l e
->
C = Colu M o n N ( r )
he6.
->
C = Epi,,,(r,Q)u E p i N ( r , r )
EpiN(r,n) = {el!). (c)
Nr i s N o - s i m p l e
=>
C = G0
and
.
(if
NEW,,
84 PRIMITIVE NEAR-RINGS
108
Proof. ( a ) follows from the f a c t t h a t h s C : Ker h so e i t h e r Ker h = Col ( t h e n h s M o n N ( r ) ) o r Ker h = r ( t h e n h = 0 ) . We may assume t h a t I f hEC A ] yeel: h(y)sel then h 9 6 , so h s M o n N ( r ) . Now h ( r ) = h(Ny) = Nh(y) = r . ( b ) V hsC: Im h = r .
Im h sN
r,
so e i t h e r
Im h = R
SIN
r
r,
+ Iol.
or
( c ) f o l l o w s from ( b ) . i n which We a r e mainly i n t e r e s t e d i n t h e c a s e t h a t C = G o , e v e r y n o n - z e r o N-endomorphism o f r i s an N-automorphism. 4.17 PROPOSITION (Betsch ( 6 ) ) .
( a ) G i s f i x e d - p o i n t - f r e e ( 1 . 4 ( b ) ) on e l . (b)
Nr i s s i m p l e t h e n
If
P r o o f . ( a ) Assume t h a t f o r
'd
6Er
3
= ny = 6 .
( b ) ->:
Then
nEN:
So
6 = ny.
C = Go <-> qEG
Then
and
1
A
yeel
r : r iN A. ~ ( y )= y .
g ( 6 ) = g(ny) = ng(y> =
g = id.
Assume t h a t h i s an N-isomorphism r + b c N h s C = G 0 c {6lu A u t N ( r ) , a c o n t r a d i c t i o n .
r.
I f hEC, h 6 then Ker h 9 r . s o Ker h = I o l . ?r T h e r e f o r e h i s a monomorphism and r Irn h . So Im h = r , a n d h s A u t N ( r ) . <=-:
-
4.18 C O R O L L A R Y ( B e t s c h ( 6 ) ) . I f Nr i s o f t y p e 1 o r i f i s s i m p l e a n d f i n i t e t h e n C = Go.
Nr
P r o o f . I f N r i s of t y p e 1 t h e n Nr i s s i m p l e . Assume t h a t h i s a n N-isomorphism r * A < N r. R e p r e s e n t r a s r = Ny and c a l l h ( y ) = : 6 . N6 = Nh(y) = h(Ny) = = h(r) = A. I f & E e l t h e n N6 = r . s o r = A , a c o n t r a d i c t i o n . I f beeo t h e n N6 = R = I o l , s o A = { o l a n d t h e r e f o r e r = C o l , which a q a i n i s a c o n t r a d i c t i o n .
109
4a General
Now a p p l y 4.15(d) a n d 4 . 1 7 ( b ) .
Nr
If 4.19 NOTATION
i s s i m p l e and f i n i t e ,
For
apply 4.17(bj.
y , 6 ~ ~ we r define (o:y)N
<->
6:
y
=
(0:a)
0 ?r
y
4.20
%
No
;
Go(y) = G o ( 6 ) .
6 : <->
REMARKS ( B e t s c h ( 5 ) ) .
(a)
%,?
( b ) The e q u i v a l e n c e c l a s s e s o f of (c)
r.
are equivalence relations i n
r.
on
Go
y,6Er:
=> (0:y)
%
y%6 -> NO
=
(for
p 6
(o:q(y))
The r e a s o n f o r d z f i n i n g
% %,%
?
are exactly the o r b i t s
= (0:s)
NO
via
3
%
y%6 ->
No
0
=> y a ) .
i n s t e a d o f N stems
f r o m 4 . 1 3 ( c ) : i n t h e f r e q u e n t case t h a t R = o t h e r w i s e be t h e a l l - r e l a t i o n i n a n y c a s e . 4.21 P R O P O S I T I O N (Betsch ( 6 ) ) . % t h e n ?, a n d % c o i n c i d e o n Proof.
If
nly
y%6
If
h : 'I ny
+
+
r
n6
= ( o : 6 ) =>
N = No
nl,n2~N n16
i s well defined.
t o be an N-automorphism,
Now
would
el.
= n 2 y ==> nl-n2E(o:y)
Therefore
r, ?
Nr i s u n i t a r y a n d
then f o r a l l
(y,6Ee1)
g ( y ) = 6 ==>
gEGo:
so
h ( y ) = h ( 1 y ) = 1 6 = 6,
= n26.
h turns out
hEG. hence
y
5
6.
84 PRIMITIVE NEAR-RINGS
110
3 . ) INDEPENDENCE A N D D E N S I T Y
An a p p r o p r i a t e frame f o r o u r
n e x t c o n s i d e r a t i o n s i s g i v e n by
4 . 2 2 D E F I N I T I O N ( M l i t t ( 9 ) ) . L e t M be a n a r b i t r a r y s e t a n d t h e s e t of a l l f i n i t e s u b s e t s o f M . A map f(M) r : f ( M ) + INo i s c a l l e d a r a n k map i f
(a) r(0) = 0 (b)
1
(c)
ti F€f!(M)
FE$(M)
mEM:
ti
r(F u E m l )
m,ncM:
= r(F)+a
with
aEI0,l)
[ r ( F u ( m l ) = r ( F W I n 1 ) = r ( F ) ->
r ( F w { m , n I ) = r(F)].
->
F i s then c a l l e d r-independent i f
r(F) = IF[.
4.23 R E M A R K
With r e s p e c t t o r - i n d e p e n d e n c e , S t e i n i t z ' s t h e o r e m i s f u l f i l l e d ( s e e A . Kertesz, "On independent s e t s of
e l e m e n t s i n a l g e b r a " , Acta S c i . M a t h . 260 - 2 6 9 ) . S e e a l s o K a a r l i ( 2 ) .
(Szesed) 21, 1960,
4.24 EXAMPLES
r ( F ) : = I F [ . Then r i s a r a n k f u n c t i o n a n d every ( f i n i t e ) subset i s r-independent.
( a ) Define
( b ) Take a v e c t o r s p a c e M o v e r a f i e l d K .
Set r(F): = = dim L ( F ) ( l i n e a r h u l l ) . r i s a rank function and r-independence i s j u s t l i n e a r independence.
( c ) Take a n N - g r o u p r a n d d e f i n e f o r e a c h
r(@) a s t h e number o f non % - e q u i v a l e n t g e n e r a t o r s ( i . e . r ( Q ) = l @ n 6 1 / Q l ) . Then r i s a r a n k f u n c t i o n a n d #
= {y l y . . . , y n l
i s r-independent
if
oEf(I')
#eel
tr i S j : v l h j . This independence i s c a l l e d %-independence. The same c a n be d o n e f o r 5.
and
111
4a General
In the theory of rings each primitive r i n g R i s isomorphic t o a " d e n s e " s u b r i n g T'? o f a r i n g HomD(I',I') f o r some i r r e d u c i b l e R-module I' a n d w i t h D = HomR(I',I') ( t h e c e n t r a l i z e r ) making Dr i n t o a v e c t o r s p a c e ( s e e ( N . J a c o b s o n ) , p . 26 3 1 ) . D e n s i t y means h e r e ( i n o u r n o t a t i o n ) t h a t SEN ( y l , ...,y s l lin. indep. i n r 1 61 6 5 ~ r 3 FeR j i E : ( l , sl: r(yi) = 6i.
-
...,
,...,
( I t i s c l e a r t h a t o n l y v a l u e s o f i n d e p e n d e n t e l e m e n t s c a n be arbitrarily prescribed.) We a r e g o i n g t o p r o v e s i m i l a r t h e o r e m s f o r n e a r - r i n g s . B u t b e f o r e d o i n g s o we h a v e t o t a k e a l o o k a t t h e d e n s i t y c o n c e p t ( s e e a l s o Adler ( 1 ) and Ramakotaiah-Rao ( I ) ) . 4 . 2 5 N O T A T I O N L e t M be a s u b s e t o f some a t o p o l o g y i n M as i n B e t s c h ( 7 ) :
We i n t r o d u c e
M(T).
I f mcM a n d y c r , d e f i n e S ( m , y ) : and q : = fS(m,y)[mcM A y E r 1 .
= Im'EM(m'(y)
= m(y)l
4 . 2 6 P R O P O S I T I O N ( B e t s c h (7),(11)). (a)
yis
t h e s u b b a s e o f some t o p o l o g y t o p o l o g y " ) on M.
i s dense i n M
( b ) NEM
<->
V
SEIN
tf m e M
w.r.t.
7
yl, . . . , y
7
(the "finite
<==>
s ~ r3
nEN
V
ie(l,
...,$ 1 :
: n ( q ) = m(v+
P r o o f . s t r a i g h t f o r w a r d and hence o m i t t e d . I n a l l t h a t f o l l o w s , " d e n s i t y " means " d e n s i t y w i t h r e s p e c t t o o f 4.26". 4.27 REMARKS ( a ) I f M and N a r e s u b n e a r - r i n g s o f M ( r ) t h e n i t i s easy t o see t h a t No i s d e n s e i n Mo i f f No+Mc(r) i s d e n s e i n Mo+Mc(r). N o t e t h a t N o t M c ( r ) a n d M o + M c ( r ) a r e no n e a r - r i n g s i n g e n e r a l ( s e e 4 . 5 3 ( e ) ) , e x c e p t i n some i m p o r t a n t s p e c i a l c a s e s . ( S e e 4 . 5 4 and 4 . 6 0 . )
112
$ 4 PRIMITIVE NEAR-RINGS
( b ) If N i s d e n s e i n i n Mo.
M then
9 {id)
( c ) Observe t h a t i f H
r
automorphism group of
tf
b
mEMH(r)
= NnMo(r)
i s dense
i s a fixed-point-free
MH(r) r
then
(since
Mo(l')
h(m(o)) = m(o)).
hEH:
If H = { i d )
No:
then
M(r).
MH(r) =
( d ) We w i l l b e m a i n l y i n t e r e s t e d i n n e a r - r i n g s w h i c h a r e d e n s e i n M (r) a n d 77 (I-):= M (r)+Mc(r) G: G: G: ( 4 . 5 2 and o t h e r s ; . 4 . 2 8 T H E O R E M ( R a m a k o t a i a h (2), B e t s c h ( 7 ) ) .
L e t H be a f i x e d -
r.
p o i n t - f r e e g r o u p o f a u t o m o r p h i s m s o f some g r o u p
(a)
W
v
A
(b)
ti
YET*
3
6Er
mEMH(r):
~ ' E P \ H ~ m: ( y ' ) =
tf S E N b
y1 y
0).
., Y ~ E ~ *
. .
,
Hyi
6 1 y . . . y 6 s ~ r 3 mEMH(r) (c) If
H
6
<->
b
SEN
tf
{id),
W
Mc(r)
(d) I f
3
61y
Proof.
W
nEN
c RH(r),
N
v
V SEN
<->
v
E
iE{l,...,sI:
, Hri
Y~,....Y,E~*
Hyi
...,d S € r 3
nEN
Hyj
tj i c { l , . . . , s I :
I n a n y c a s e we may assume t h a t RH(r) = MH(r)
otherwise (a)
aEHy
-fl
haEH:
Define
mEM(r)
by
a = h,(y)
m(y)
= 6
and
0
mEMH(r);
d e t e r m i n e d by t h e c o n d i t i o n s ( v y ' ~ r \ H y : m(y')
9j
= bi.
HH(r) <-> for
i
+j
= 6i.
{id),
for
(since H i s fixed-
haV)
m(a):= I
clearly
n(yi)
n(yi) H
i
for
M(r).
point-free).
A
=+
= 6i.
m(yi)
Hyj
=/=
N i s dense i n
ylY...,ysEr,
j
M H ( r ) <->
... ,s?:
i E { l ,
i
for
Hyj
i s dense i n
NrMH(r)
61y...y6s~r
= 6 A
(m(y)
= 0).
m(y)
aEHY
.
Then a&HY m i s uniquely = 6 A
113
4a General
( b ) D e f i n e maps
mi€MH(r)
b
= o
mi(y')
y'4Hyi:
m: = m l +
Then
with
...+m S
w i l l do t h e j o b .
By ( b ) a n d 4 . 2 6 ( b ) .
<-:
If
Hyi
p
Hyj
If
Hyi
= Hyj,
V
n(yj)
and
= tii
(as i n ( a ) ) .
( c ) =+:
->
mi(yi)
for
i
9 b
mEMH(r)
the result i s clear.
j,
= m(yi)
nEN: n ( y i )
a>
and t h e r e s u l t f o l l o w s a g a i n f r o m
= m(yj)
4.26( b ) . ( d ) +: By 4.27,
I f one
( s a y yl)
yi
= bi-6
V
fulfills
ncEMc(r) Take
= 61.
...,sl.
for
i ~ { 2 ,
ic{l,.,.,s):
Two o r m o r e
take
= 0,
map w h i c h i s c o n s t a n t no(yi)
n(yi)
=
MH(r).
t o be t h e
n0ENo
with n: = notn
Then
C
= tii.
c a n n o t be z e r o .
yi
(R,,(r))o
i s dense i n
No
If all
yi
the
0,
r e s u l t follows from ( c ) . <==: I f
and
yl,
SEN,
... , y S c r * define
61,...,6s~r,
Then
3
n(o) =
nEN 0,
iE{l,
nENo
so by 4 . 2 7 a )
,
Y ~ + ~= :o
..., s + l l :
and b y ( c ) ,
M i s dense i n
4.29 THEOREM ( B e t s c h ( 7 ) ) .
No
i
and
6s+1:
=
= tii.
H 5 Aut(T)
and
r
Then t h e f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t :
(a)
i s discrete i n
RH(r).
(b)
i s discrete i n
MH(r).
( c ) H h a s o n l y f i n i t e l y many o r b i t s o n
( i n t h i s case, H
+ {id}.
( a ) ->
Then
r
t r i v i a l l y hold f o r
i s finite). since
MH(r),
as i n
r. H = {id).
S o we assume t h a t
MH(r)Em0.
(b): Trivial,
0.
Because o f
i s dense i n
4.26.
Proof. Again the r e s u l t s
j
g,,(r).
rev,
Let
n(yi)
for
Hyj
Hyi
MH(r) f
FIH(r).
§4 PRIMITIVE NEAR-RINGS
I14 ( b ) ->
( c ) : Assume t h a t H h a s i q f i n i t e l y many o r b i t s
r.
on
3
Then
If
msMH(r) a n d a n e i g h t o u r h o o d U o f m.
Take
3
SEN
y1
= HyJ
Hyi
assume t h a t
,..., y S ~ F :U
then Hyi
S(m,yi)
+ Hyj
for
S
3
fl
i = l
S(m,yi). So we w i l l
= S(m,yj).
i
p
j.
S i n c e H has i n f i n i t e l y many o r b i t s ,
3
y S t l E r \ ( { o l u Hyl u
\
Then
Then
ml:
=
= m.I
P0
If m ( y S t 1 ) = so
m2
Anyhow,
m2
0
0
3
Hy,).
\y'
eiE:MH(I')
...t e s )
m (el+
and
ml
If m ( Y s t l ) m1
...,s t l l
i t C l ,
Define
...
and
E
fl S(m,yi) i=l
then
ml(Ys+l)
then
m2(yStl) = Y
dp m . U c o n t a i n s an e l e m e n t
= 0
+m
...,s t l l :
= mltestl.
m2:
S
are
jE.Il,
5 U.
=I= m(yst,), ~
+
and
+ 0 ~= m
so
(
~
c a n n o t be
discrete. ( c ) ->
(a):
I f H h a s o n l y f i n i t e l y many o r b i t s o n
t h e n each element o f
MH(r)
and o f
flH(r) i s
u n i q u e l y d e t e r m i n e d b y i t s e f f e c t o n f i n i t e l y many s u i t a b l e elements o f So
7is
discrete.
r.
r
~
~
115
4b 0-primitive near-rings
b l 0-PRIMITIVE N E A R - R I N G S
Now we s h a l l p r o v e a " d e n s i t y - l i k e "
s t r u c t u r e theorem f o r
0 - p r i m i t i v e n e a r - r i n g s , Gle s t a r t w i t h z e r o - s y m m e t r i c o n e s . We may a s ~ u m e ( 1 . 4 6 ) t h a t i f N i s 0 - p r i m i t i v e o n r t h e n N Q M ( T ) . G e n e r a l i z a t i o n s c a n b e f o u n d i n M l i t z (4),(3),(12) a n d K a a r l i ( 6 ) .
4 . 3 0 T H E O R E M ( B e t s c h ( 6 ) ) . L e t NewO be 0 - p r i m i t i v e o n r . I f N i s a r i n g t h e n N i s a p r i m i t i v e r i n o on t h e N-module
r and Jacobson's d e n s i t y theorem i s a p p l i c a b l e . I f N i s a n o n - r i n g t h e n we q e t a k i n d o f a d e n s i t y property:
...,y , ~ r , i c { l , ..., s ? : nyi
(D): \ scIN
3
ncN
Proof.
61,...,6s~r
6i.
N i s a r i n g we o n l y h a v e t o a p p l y 4 . 8 .
If
Now l e t l e t
V
%-indep.
yl,
N
s be
I n t h e t e r m i n o l o g y o f (D),
be a n o n - r i n g . > 1
and f o r
v
Lemma.
t
...,S I : fl
kc{ttl,
Vtc{l,
...,s - 1 1
tc{l,
be t h e statement
i = l
...,s - 1 1 :
(0:Yi)
let
$ (0:Yk)
S(t)
,
S(t).
P r o o f . By i n d u c t i o n o n t . Since f o r
(o:y)
YEel
V
i d e a l o f N,
i,jE{l,
i s a maximal l e f t
...,s l :
(o:yi)~
( o : y . ) = ( o : y . ) => yi%yj -=> i = j . J 1 J Particularly: S(1): k~(2, s l : (o:yl)$
E ( o : ~ . ) =>
v
...,
9(O:Yk). Now assume Then Since
S(t),
t
fl
s 2 3
(O:Yi)$(O:Yk) i=l (o:yk)
and
and
...,sl.
k~ftt2,
(o:Yt+l)$(o:Yk)*
i s maximal,
t
[I
(o:yi)+(o:Yk)
i=1
= (o:Yt+l)+(o:Yk)
= N*
$ 4 PRIMITIVE NEAR-RINGS
116
Since N i s not a r i n q , by 3 . 4 ( i ) , $(o:yk) than S f t t l ) .
t
n
n (o:yttl)+
(o:vi)
i=l
which i s n o t h i n q e l s e
Now r e t u r n t o t h e p r o o f o f 4.28 and l e t yl, . . . , y , , be a s i n (D). A g a l n we u s e i n d u c t i o n o n 61,.,.,6 tc{l,
S
...,s l .
t = 1
If
then
b
Now assume t h a t
b
iE{1,
..., t l :
By t h e lemma,
3
Therefore
Now we t a k e
: nttlyi
t
n
i=l
Since
(o:vi)$(o:y
LYttl
RcL: Lyt+l
= 6i,
nlyl
3
= 61.
ntEN
= 6i.
ntyi
nttl:
nlaN:
tE{l,,..,s-ll
L: =
+ {ol.
Lyttl
3
ylEel
$N
t+l),
hence
(3*4(a)),
= 6ttl-nt6t+l
=
LYttl
r.
.
and g e t \1 i E { l , and t h e p r o o f i s c o m p l e t e . = Ltnt
...,t t l l :
4.31 R E M A R K S
(a)
(D)
i s no " r e a l " d e n s i t y p r o p e r t y s i n c e t h e r e i s no
n e a r - r i n g i n s i g h t i n which f i n i t e topology). ( b ) From
and
(D) i t
N i s dense ( w . r . t t h e
+)
f o l l o w s (Ramakotaiah ( 2 ) ) t h a t ,
if
SEIN
a r e %-independent,
yl,...,yscr
?r
i=l
= W(Yi))). ( c ) The c o n t e n t o f 1Bli
= 1,
(0)
m i g h t be v e r y t h i n :
i f e.g.
(D) i s t r i v i a l . So i t i s n o t t o o s u r p r i s i n g
t h a t t h e c o n v e r s e o f 4.28 does n o t h o l d :
+)
( B e t s c h ) : If o n e c h a n g e s o f k.25 t o Y':={S(m,y)ln&:Mn A y€e,(r)] t h e n o n e g e t s a " r e a l " d e n s i t y t h e o r e m w.r.t. t h e r e s u l t i n g c o a r s e r topology.See also 9.230.
4b 0-primitive near-rings
117
L e t N be t h e n o n - r i n q {fEMo(iZ4)lf(2)E{0,21Af(3) 3f(l)I. In z4, O 1 = { 1 , 3 1 , 1 % 3 %( D ) i s f u l f i l l e d , b u t I0,21 QN Z4, so Z4 i s n o t s i m p l e a n d t h e r e f o r e N i s n o t 0 - p r i m i t i v e on Z 4 .
( d ) ( D ) i s equivalent t o t h e following property: (0’):
3
v
SEN
iEI1,
nEN
yl,. . . , y ,cr,
%-indep.
mE:M(r)
.,.,sI: nyi = m ( y i ) .
Now we t u r n t o a r b i t r a r y n e a r - r i n n s . 4.32 T H E O R E M
( a ) Let N be 0 - p r i m i t i v e o n r. Case 1: Nor -/= { 0 1 . Then N o i s 0 - p r i m i t i v e on so 4.30 i s applicable ( f o r r ) , and NC Case 2 :
5
NO
Mc(r).
Nor
(01.
r,
Then
Mc(r)
N =
and
r
i s a
non-zero simple g r o u p i s 0 - p r i m i t i v e on r a n d ( b ) Conversely, i f e i t h e r N o N, E Mc(r) o r i f N = Mc(r) where r (01 i s s i m p l e t h e n N i s 0 - p r i m i t i v e on Proof.
( a ) Anyhow,
If and
Nor
I01
e
N,
Mc(r).
then NO
No
r.
r
i s 0 - p r i m i t i v e on
i s o f t y p e 0 by 3 . 1 8 ( a )
r
(3.15(a)).
“by f a i t h ” , s o R = r a n d N = N, = Mc(r) by 1 . 5 0 ( b ) . S i n c e Ncr i s s i m p l e i f f r i s s i m p l e , ( a ) i s shown ( o b s e r v e t h a t Ncr Col!). If
Nor
=
(01,
No
= {01
+
( b ) A g a i n b y 3 . 1 8 ( t h i s t i m e by ( b ) ) , i f
N r i s o f t y p e 0 . S i n c e NccMc(r), N o a n d N, ( a n d hence N ) a c t f a i t h f u l l y o n r , s o N i s 0 - p r i m i t i v e o n I’. I f N = Mc(r), r $. Iol and s i m p l e , t h e r e s u l t i s clear.
0 - p r i m i t i v e then
118
4.33
§ 4 PRIMITIVE NEAR-RINGS
R E M A RK
(D) w o u l d
and i n
No
= (0:6)
r,
( i n
if
n o t n e c e s s a r i l y mean t h e same i n would be d e f i n e d by
%
Nr
(o:y)
y%fi:<->
=
il). Cf. 4 . 1 9 .
ideal i s a
4 . 3 4 THEOREM ( R a m a k o t a i a h ( 1 ) ) . E a c h 0 - p r i m i t i v e prime i d e a l
dp N.
P r o o f . L e t I b e a 0 - p r i m i t i v e i d e a l o f N. L e t Nr b e o f t y p e 0 w i t h g e n e r a t o r yo s u c h t h a t I = ( 0 : r ) (4.3). Assume t h a t
Ji $ (o:r),
J1J2
Jir SO
9 N: J1J2tI
= JiNy
(o:r) = I,
=
c Jiyo
+ {ol.
= Jiyo
Jiyo
0
r.
Now
J1 $ I
A 5
A J2$I.
Jir.
Since
By 3 . 4 ( a ) ,
r,
= Jlr =
JlJ2r
a contradiction.
R E MA R K I n 5 . 4 0 we w i l l s e e t h a t t h e c o n v e r s e o f 4 . 3 4
holds i f 4.36
r.
dN
Jiyo
4.35
J1,J2
i ~ { 1 , 2 1 , Jir
For
so
3
N = No
h a s t h e DCCN.
THEOREM ( R a m a k o t a i a h ( 1 ) ) . E v e r y m a x i m a l m o d u l a r i d e a l
I of
Nsno
Proof.
i s a 0 - p r i m i t i v e one.
L e t I b e a m o d u l a r m a x i m a l i d e a l . By 3 . 2 2 ,
I is
c o n t a i n e d i n a modular maximal l e f t i d e a l L . Since (L:N)
i s the largest ideal of N contained i n L
( b y 3.25), we g e t
by 3.21(a). By 4 . 3 ( c ) ,
I?(L:N)
and
(L:N)
By t h e m a x i m a l i t y o f I,
I i s 0-primitive,
i s modular
I = (L:N).
s i n c e b y 3.29 L i s
0-modul a r . By t h e w a y ,
if N i s 0 - p r i m i t i v e on
n o t necessarily simple (K.
r
a n d Ny = : A t h e n A i s
Kaarli).
F o r t h e r e s t o f t h i s s e c t i o n , we g i v e a d e s c r i p t i o n o f a c l a s s o f 0 - p r i m i t i v e near-rinqs which are n o t 1 - p r i m i t i v e . This d i s c u s s i o n i s d u e t o H o l c o m b e (5), w h e r e t h e p r o o f s c a n b e found,
too.
119
4b 0-primitive near-rings
4.37
4.38 DEFINITION I f
:'G
=
EFr, H < A u t ( T ) ,
(r,+)E(a,
t h e t r i p l e (T,B,H)
be the s e t of
r\el
NOTATION I f N r E N q , l e t A : = generators". If A sN r , let (cf. 3.14(a)!).
-
"non-
AutN/(o:A) ( A )
H ( B ) ~ B we , ,-all
c o m p a t i b l e i f a t l e a s t one o f t h e
following conditions i s satisfied:
r.
( a ) B i s no n o r m a l s u b g r o u p o f
3 YEAB 2 (3 h ' c H 3
(b) (c)
(3
A
4.39
BEB
tl
y~r\B
+
hEH : y+B h(y). R E B : y+B = h ' ( y ) ) h
1
Y ' E ~ \ B: h ' ( y ' ) - y ' & B ) .
be 0 - p r i m i t i v e on r , N a n o n - r i n o w i t h NE% i d e n t i t y a n d DCCL, a n d l e t A ( a s i n 4 . 3 7 ) b e an N - s u b g r o u p
THEOREM L e t
of
r
Then
such t h a t
i s not f a i t h f u l , b u t o f type 2.
NA
N i s n o t 1 - p r i m i t i v e on A,
many o r b i t s o n el,
G ( 4 . 1 0 ( b ) ) has f i n i t e l y
i s c o m p a t i b l e and
(T,A,G)
(where A i s a f i n i t e dimensional v e c t o r space o v e r the d i v i s i o n r i n g
GAvta)).
Conversely:
4.40
THEOREM L e t subgroup.
r
Let
b e an a d d i t i v e g r o u p and A be a n o n - z e r o
GA
be a r e g u l a r group o f
automorphisms
o f A w h i c h h a s o n l y f i n i t e l y many o r b i t s o n A. L e t H b e a subgroup o f (a)
( b ) each
Aut
(r,+)
such t h a t
i s compatible.
(r,A,H) hEH
i s r e g u l a r on
AA.
( c ) H h a s o n l y f i n i t e l y many o r b i t s o n (d)
hEH: h/,EG*.
Then
N = IfEMHu{a) ( r ) I f / A E M G A ( A ) l
0-primitive, a n d t h e DCCL.
b u t n o t 1 - p r i m i t i v e on
nA.
i s zerosymmetric,
r,
has an i d e n t i t y
5 4 PRIMITIVE NEAR-RINGS
120
r9
I f moreover
and A i s a f i n i t e d i m e n s i o n a l v e c t o r
A
tJ
s p a c e o v e r some d i v i s i o n r i n g D a n d i f then
N = If€M,,"~,)(r)lf/,EEndD(r))
r,
b u t n o t 1 - p r i m i ti ve on
"(
NE
hEH:
h/AED
i s also 0-primitive,
no,?i!z 3, , a n d
moreover
i s a ring.
o :A)
4 . 4 1 R E M A R K See a l s o H o l c o m b e ( 4 ) f o r t h e m o r e q e n e r a l c a s e t h a t A i s o n l y a f i n i t e u n i o n o f N-subgroups o f t y p e 2 w i t h zero intersection. 4.42
t h e n i n t h e n o n - r i n g case o f 4.30
GA = { i d )
REMARK I f
we g e t n e a r - r i n g s o f t h e f o r m N = ( f c M o ( r ) l f ( A ) F A ] ( s e e e . g . N o i n 3 . 8 ) . C f . R a m a k o t a i a h - R a o (1),(3),(4). Conversely,
(r,t)
if
i s a f i n i t e group and A a non-
t r i v i a l subgroup then
N:= { f E M o ( r ) l f ( A ) F A }
i s a finite
n e a r - r i n g w i t h i d e n t i t y , z e r o - s y m m e t r i c a n d 0-, b u t n o t 1 - p r i m i t i v e o n r. A i s j u s t t h e s e t o f n o n - g e n e r a t o r s and i s an N-subgroup s u c h t h a t N / ( o : A ) i s a non-ring if
lAI
> 2.
C\
1 - P R I M I T 1 V E N E A R - R I FIGS
Now l e t N b e 1 - p r i m i t i v e o n
r r
r.
C = Go
Then
(by 4 . 1 8 ) ,
i s n o t N-isomorphic t o a p r o p e r subgroup (4.17(bl),
= eouel
R = Col
(by 4.15(d)),
+
t~ L
Ak N , L EOI 3 Y E r : We s t i l l a s s u m e t h a t N E 4.43
THEOREM
Ly =
M(r).
or
r
R =
r
(3.2)
and
(by 3.4(a):.
r . Then Nor f ( 0 ) A R = r . T h e n N o r , Nc = Mc(r) a n d e l = r .
(a) L e t N be 1 - p r i m i t i v e on C a s e 1:
on
If
No
i s a r i n g then
i s 1-primitive
N i s dense i n
Maff(r)
where r i s a v e c t o r space o v e r t h e d i v i s i o n r i n g D : = HomN (F,r). 0
121
4c 1 -primitive near-rings
If
No i s n o t a r i n g t h e n applicable.
(D)
Case 2 : Nor ( 0 1 A R = ( 0 1 . Then m i t i v e on r a n d 4.30 h o l d s . Case 3: Nor = { o l . Then N = N, a s i m p l e g r o u p dp (01.
=
o f 4.30 i s
N = No
Mc(r)
i s 1-priand
r
is
( b ) C o n v e r s e l y , i f a n e a r - r i n g NEM(r) i s such t h a t N o i s 1 - p r i m i t i v e on r w i t h Nce(IO1, M c ( r ) l o r i f N = Mc(r) (r { o } a n d s i m p l e ) then N i s 1 - p r i m i t i v e on r .
+
+
P r o o f . ( a ) I f Nor (01, N o i s 1 - p r i m i t i v e on r by 3 . 1 8 ( a ) . S i n c e e a c h s t r o n g l y monogenic N-group has e i t h e r R = ( 0 1 o r n = r, t h e r e s t f o l l o w s from 1 . 5 0 , 3.9, 3 . 1 5 ( a ) , 4 . 2 7 ( a ) a n d 4 . 3 2 . ( b ) I f No i s 1 - p r i m i t i v e on r and N, = {O) o r 'L N, Mc(r) t h e n e i t h e r R = ( 0 1 o r R = r ( 1 . 5 0 ) , s o N i s 1 - p r i m i t i v e on r by 3 . 1 8 ( b ) and 3 . 1 5 ( a ) . 'L If N Mc(r), r s i m p l e and (01, then c l e a r l y
-
-
N i s 1 - p r i m i t i v e on
r.
+
4.44 R E M A R K 4.43 i s t h e main r e a s o n f o r d e f i n i n g " s t r o n g l y monogenic N-groups r" as i n 3 . l ( b ) a n d n o t by t h e c o n d i t i o n s "monogenic" and ycr: (Ny = R v Ny r)", f o r 4 . 4 3 would n o t be t r u e i n t h i s c a s e : Take 'l = H8, N o : = { f ~ M ~ ( r ) l f ( =P )f ( 6 ) c 1 0 , 2 , 4 , 6 1 A N,: = {fcMc(r)lf(0)c{0,2,4,611. A f ( 4 ) ~ { 0 , 4 ) ) and
'v
T h e n one can show t h a t N : = N o t N c i s a subnear-ring of M ( r ) enjoying the following properties:
Nr
and NO
r
a r e f a i t h f u l , s i m p l e a n d monogenic. Moreover,
ycr: (Ny = R = { 0 , 2 , 4 , 6 ) v Ny = r). B u t (01 f R r, and N o i s n o t 1 - p r i m i t i v e on r ( i t i s n o t e v e n t r u e t h a t f o r a l l y c r Noy i s e i t h e r = { o l , = R o r = r, s i n c e N04 = { 0 , 4 1 ) .
122
8 4 PRIMITIVE NEAR-RINGS
From 4 . 3 0 a n d 4.45
r
= e0u
el
we g e t w i t h a s t r a i q h t f o r w a r d p r o o f
N€TIO
THEOREM L e t t h e n o n - r i n a
r
be 1 - p r i m i t i v e on
but
without %-equivalent generators.
N
Then
i s dense i n t h e n e a r - r i n q
€370
For 1-primitive near-rinqs
{ f c M o ( r ) \ f ( e o ) = {oil.
w i t h D C C we g e t a w h o l e b u n c h
o f i m p o r t a n t r e s u l t s ( c f . Rarnakotaiah ( 3 ) , B e t s c h ( 1 0 ) ) : 4 . 4 6 THEOREM ( B e t s c h (3)).
Let
e n d o w e d w i t h t h e DCCL.
NEWO
be 1 - p r i m i t i v e on 7 and
Then
( a ) T h e r e a r e o n l y f i n i t e l y many % - e q u i v a l e n c e c l a s s e s i f N i s a non-rinq. (b)
3
S E N: ,,N
s. =
N-isomorphic
1 Li,
r)
l e f t i d e a l s and N-groups o f t y p e 1 ( s o
2.50 i s a p p l i c a b l e ! ) ; s =
f i n i t e l y many p a i r w i s e ( t o
Li
i = l
i f N i s a non-rinq then
Ir/%l-i.
( c ) A l l N - g r o u p s o f t y p e 0 a r e N - i s o m o r p h i c a n d o f t y p e 1. (d) N contains a r i g h t i d e n t i t y (not necessarily two-sided). (e)
N is s i m p l e .
( f ) N i s e i t h e r 2 - p r i m i t i v e o n 'I o r t h e r e i s n o N - g r o u p
o f t y p e 2, Proof.
If N i s a ring,
(b)
-
( f) are e i t h e r well-known
o r t r i v i a l . So we w i l l a s s u m e t h a t
No
i s a non-rinq.
( a ) S u p p o s e t h a t t h e r e a r e i n f i n i t e l y many - - e q u i v a l e n c e
v,,vl.y2,...
classes w i t h representatives assume t h a t
i 2 1,
y o ~ e o . Then
hence
(o:yo)=(o:y1)=(o:~y1,y2))3'. d i c t i o n t o t h e DCCL.
( b ) Now l e t
Y
~
~
representatives of
YOEeOl
yl,
(o:yo)
y1yy21...~f31.
Y
.~'yS~
= N
p
.
We may
(o:yi)
for
So b y ( D ) o f 4 . 3 0
.
which i s a contra-
. b e. a c o m p l e t e s y s t e m o f
t h e %-equivalence classes w i t h
...,y S c e 1 .
Then
n
S
( 0 : ~ ~= ) I 0 1
i= 1
,
but
123
4c I-primitive near-rings
minimal l e f t i d e a l s . Now a p p l y 2 . 5 0 ( p )
tf
j E { l ,
...,sl:
to get
N =
'.1 L
Since
r
by 3 . 1 0 .
j = lj * ,-b
L j -N
L.$(o:Y~), J
( c ) Holds by t h e p r o o f o f ( b ) and 3 . 1 1 ( a ) .
N contains a riqht identity
( d ) By ( b ) a n d 3 . 2 7 ( d ) ,
N1 e. sided. (e)
o f 3.8
shows t h a t e i s n o t n e c e s s a r i l y t w o -
3
If I Q N,
L j n I = C O l .
minimal, IE(0:L.) J
. . . ,s l :
jc(1,
=
{O}
But %
(for
L j -N
LjL)I.
Lj
Since
ILjcInL
F), w h e n c e
j
=(O), I = {Ol.
is SO
( f ) By 4 . 7 o r b y ( c ) .
Note t h a t 4.46(a) i s not v a l i d f o r r i n g s : I f 2 s p a c e IR , c o n s i d e r e d a s a n H o m ( r , r ) - m o d u l e , (xELP)
are pairwise inequivalent w.r.t.
n e a r - r i n g which i s p r i m i t i v e on 4.47
=/=
i s the vector all
(1,x)
Hom(r,T)
i s a
a n d h a s t h e DCCL.
N contains a l e f t identity;
COROLLARY N E ' ~ ' ) ~ . D C C N , PI
r
%,
r
I 9 N,
{Ol. T h e n
(a) N !s
1 - p r i m i t i v e c=+
N i s 2 - p r i m i t i v e <->
N i s simple.
( b ) I i s 1 - p r i m i t i v e c->
I i s 2 - p r i m i t i v e <->
I i s maximal.
Proof.
( a ) By 3 . 4 ( c )
and 3.7(c),
2-primitivity coincide.
N i s simple then
If
by 3.4(b)
1 - p r i m i t i v i t y and
I n t h i s case,
I = N
N i s simple.
i s a m i n i m a l i d e a l and
a n d Lemma 3 i n t h e p r o o f o f 3 . 5 4
I : = N; t h e n
A =
r)
(with
N has a f a i t h f u l N-group
Nr
o f t y p e 1, s o N i s 1 - p r i m i t i v e . (b)
follows from (a).
K a a r l i ( 2 ) showed t h a t if N = No i s s i m p l e a n d U i s a m a x i m a l N subgroup of
N w i t h NU
{O} t h e n N i s I - p r i m i t i v e .
K a a r l i ( 4 ) and A d l e r ( 1 ) .
Cf.
also
124
9 4 PRIMITIVE NEAR-RINGS
d ) 2-PRIMITIVE N E A R - R I N G S
A g a i n we assume t h a t i f
N i s 2 - p r i m i t i v e on r then
NCM(r).
1 . ) 2-PRIMITIVE N E A R - R I N G S The s t r u c t u r e o f 2 - p r i m i t i v e n e a r - r i n g s c a n b e d e s c r i b e d as follows.
4.48 THEOREM ( a ) L e t N be 2 - p r i m i t i v e on Case 1:
Nor
I01 A R
r. Then = r. Then
No
i s 2-primitive
N, = M c ( r ) a n d e l = r . No i s a r i n g t h e n N i s d e n s e i n M a f f ( r ) (as i n 4.43); i f No i s a n o n - r i n g t h e n (0) o f 4 . 3 0 i s a p p l i c a b l e ( f o r No).
on
r,
If
Nor 9 { o }
A
R = {ol.
p r i m i t i v e on
r
and 4 . 3 0
Case 2 : Case 3 :
Nor
=
(01.
Then
Then
N
= No
i s 2-
i s applicable.
N = Mc(r)
and
r
is a
c y c l i c group o f prime order. Conversely,
(b)
if
Nce{{O),Mc(r))
No
i s 2 - p r i m i t i v e on
or if
N = Mc(r)
(r
r
o f prime o r d e r ) then N i s 2 - p r i m i t i v e on The p r o o f i s s i m i l a r t o t h e o n e o f 4 . 4 3
4.49
r
a n d if I A
unless order).
I = Mc(r)
r.
and t h e r e f o r e o m i t t e d .
Betsch ( 7 ) ) . I f N i s 2 - p r i m i t i v e { 0 1 , t h e n I i s 2 - p r i m i t i v e o n r, (where r i s n o t a c y c l i c qroup o f prime
T H E O R E M ( c f . F a i n (1) a n d
on
with
a c y c l i c group
N, I
p
125
4d 2-primitive near-rings
NEQ.
( a ) We f i r s t s h o w t h i s t h e o r e m f o r
Proof.
Ir
Ev d e n t l y , Assume t h a t If
3
IA =
i s faithful.
sI r.
A
Iol
then consider
{01.
~ E A : N6
Ir =
But
Therefore
INACIA =
i s f a i t h f u l . Hence
Iol, NA =
A = {o}.
If
I A
+ {ol
N6 =
I
so
(01,
3
then again
NA
If
NA.
6fA:
r
(01, and N A =
{Ol, A
I6
.b
since
sN f , =+
whence
Io1.
16 sN r , s o 16 = Consequently A = r , f o r I6cA. T h e r e f o r e I i s 2 - p r i m i t i v e o n r. ~ ( 1 6 )= (NI)~FIG,
r. Ir
Since
r.
( b ) Now l e t N b e a r b i t r a r y . We may assume t h a t
N
No,
=/=
I s excluded.
so case 2 o f 4.48
I f N f a l l s i n t o c a s e 1, No i s 2 - p r i m i t i v e o n r . BY 2.18 , I , = ~n ri, 2 N 0 ' I f I , p (0) t h e n I, i s 2 - p r i m i t i v e o n r , hence
I i s 2 - p r i m i t i v e on r by 3.18(b). I f I, = {Ol t h e n I e N c = Mc(r). S i n c e el = r , 10 AN r . 10 = { o l i m p l i e s t h a t f o r a l l y = n o c r and f o r a l l i E I i y = ino = 0 , so I = I o ) .
Hence
10
T a k e any
3
+ Io)
and so
mceMc(T);
is1 : io =
u.
10 =
m o =:p. C
r. Because o f
10 = ,'I
= i o = p = mco = mcy, h e n c e i = mc % I = Mc(r). If 3 P E P : r Zp, I is 2 - p r i m i t i v e o n r; i f not, I i s not 2-primitive. Now
yfr:
iy
-
a n d we g e t
If N i s i n c a s e 3, 4.50
REMARK
I i s t r i v i a l l y 2 - p r i m i t i v e on
4 . 4 9 c a n n o t b e t r a n s f e r r e d t o 0- o r 1 - p r i m i t i v i t y ,
n o t even f o r f i n i t e ,
abelian, zerosymmetric near-rings.
I t i s e a s y t o show t h a t i f N i s e . g . O - p r i m i t i v e o n I 9 N t h e n I r i s f a i t h f u l and monogenic. B u t n o t necessarily simple: T a k e r: = z 8 , A : = IO,2,4,61 a n d E: = I 0 , 4 l . N:
r.
= {ffMo(r)lf(A)rA
A f(5)=f(l)
h
f(7)=f(3)),
r
and
I:= (0:A).
9 4 PRIMITIVE NEAR-RINGS
126
N i s 0 - p r i m i t i v e on
r,
but
E d1
r.
Moreover,
s t r i c t l y monogenic and I has a r i q h t i d e n t i t y . e v e n b e 0 - p r i m i t i v e o n some o t h e r g r o u p Assuming t h a t ,
take
and ( 0 : 3 ) 1
I"
Ir
is.
I cannot
=:IyA:
Ir
i n
and p u t
L : = ( 0 : ~ ; ) ~Then . L i s a maximal l e f t i d e a l o f I ( 3 . 4 ( f ) ) L (0:1)1 , L ( 0 : 3 ) 1 ( s i n c e ( 0 : 1 ) 1 and (0:3)1 c a n n o t be m a x i m a l ) . T h e r e f o r e ( o : l ) I + L = ( f l : l ) I+ L = I . b u t ( 0 : l ) n (0:3) = E ~ I c - L , s o I w o u l d h a v e t o b e a r i n g
and
by 3.4(i),
a contradiction.
So p o i n t e d o u t , N N =
A s Y.S.
I
(0:l) n(0:3).
S e e m i n g l y t h e r e i s no "smal l e r " c o u n t e r e x a m p l e t h a n t h a t above w i t h 4096 e l e m e n t s . See a l s o 5 . 1 9 ( a ) . By t h e way, case
o n e c a n u s e Z o r n ' s lemma t o show t h a t i n a n y
I 9 N (I9 IO), N w - p r i m i t i v e )
h a s some I - g r o u p s
o f t y p e v. 4 . 5 1 COROLLARY ( F a i n ( 1 ) ) . L e t P b e a 2 - p r i m i t i v e i d e a l o f NET,. L e t I b e a n o t h e r i d e a l o f N c o n t a i n i n g P . Then P i s a 2-primitive
Proof. I
9 P,
A NIP,
ideal of I . and
NIP
'Ip p I01
and
i s 2-primitive.
Since 'Ip i s 2 - p r i m i t i v e . Hence
P i s 2-primitive i n I.
2 . ) 2 - P R I M I T I V E NEAR-RINGS I.IITH IDENTITY
I n t h i s case, Also,
if
N = No
(if
=
then
%
=
%
%
NE~,)
or
el
=
r
( i fN $ % ) .
(by 4.21).
Recall that a 1-primitive near-ring cfl0 with a l e f t identity i s a l r e a d y a 2 - p r i m i t i v e one w i t h i d e n t i t y ( 4 . 6 ( b ) ) . We a r e now i n a p o s i t i o n t o g e t a " r e a l " a n d f u n d a m e n t a l d e n s i t y theorem.
4d 2-primitive near-rings
N f N,
N
(case 1
No a non-ring
No
(case 2 o f
o f 4.48)
No a r i n q
=
127
4.48)
-
Ma f f ( r )
HomD ( r ,r 1
Dr a v e c t o r s p a c e $. fol
l'fc (r)
M C ( r ) = MC(r)
Go f i x e d - p o i n t -
0
0
r
f r e e on
(b) Conversely,
e v e r y n e a r - r i n g w h i c h i s dense i n Maff(r) o r HomD(r,r) (where r i s a non-zero v e c t o r space o v e r some d i v i s i o n r i n g D ) o r d e n s e i n (r) o r
MC 0
on Proof.
(r)
r,
fixed-point-free
(Go
where Co = G
w
0
2 - p r i m i t i v e on note that
1
= MC
r
(r)
4.28(c)
and has an i d e n t i t y .
No
i s therefore I f No i s a i s not a rinq,
on
r,
= MGoV{b}
(r)
since
Y I sN r .
then
My;
0
(r)
o(r)
= M
=
Go
(D)
o f 4.30,
and 4 . 2 7 ( a ) .
Mo(r),
=/=
If
and t h e r e s u l t f o l l o w s f r o m
equal t o )
If N
No
0
Go = { i d }
in
and has an i d e n t i t y
cannot occur.
i s fixed-point-free
Go
+ {id}
Go
0
r
CvcrJs(Y) =
gE:Go:
If
If
{61.
t h e s t a t e m e n t is c l e a r .
ring,
i s 2-primitive
on 7 )
( a ) I f N i s 2 - p r i m i t i v e on
t h e n case 3 i n 4.48
mc 0
No,
then (D)
of 4.30
implies that
N
which i s t r i v i a l l y dense i n ( s i n c e
MC
0
(r)
= MCal(r)
=
Mo(r).
apply again 4.27(a).
0
i s dense
8 4 PRIMITIVE NEAR-RINGS
128
( b ) Assume now t h a t N i s d e n s e i n H o m D ( r , r ) , where i s a n o n - z e r o v e c t o r s p a c e o v e r some s k e w - f i e l d D.
r
N i s a dense s u b r i n g and t h e r e f o r e a p r i m i t i v e
Then
r i n g o n r . F r o m t h i s we d e d u c e : I f N i s dense i n M a f f ( T ) then
i s dense i n
No
s o r h a s no n o n - t r i v i a l N o - s u b g r o u p s , HomD(r,r), a n d N i s 2 - p r i m i t i v e o n r. I f N i s d e n s e i n M O(r) GO
each dense s u b n e a r - r i n g of t h a t i s t r i v i a l l y 2 - p r i m i t i v e o n r. I f G o {id} t h e n M o(r) = FIG ( r ) a n d 4.28(c)
0
GO
Nr c a n n o t c o n t a i n n o n - t r i v i a l
shows t h a t
N-subgroups. Finally i f
in
(Tic
0
N i s dense i n
(r))o
a non-trivial
MC (r)
then
0
As above,
= MC ( I - ) .
r
No
i s dense
cannot c o n t a i n
0
No-subgroup (or one can use 3 . 1 8 ( b ) ) .
4.53 REMARKS (a) I t i s n o t t r u e t h a t each 2 - p r i m i t i v e n e a r - r i n q w i t h identity, N take
MC(r): and
N: =
and
No
a non-ring,
i s dense i n
=/=
No
r
f i n i t e w i t h I i d l f G s A u t ( r ) , G f i x e d - p o i n t free Nr h a s R = r, s o
F i G ( r ) . Then
CN(r) = { i d ) ( 4 . 1 3 ( c ) ) a n d t h e r e f o r e MC(r) = M(r). But N M ( r ) , s o N c a n n o t be d e n s e i n M ( r ) b y
+
4.29.
This i s a l a t e b u t convincing reason f o r
introducing t h i s crazy s w i t c h e s down t o
No
N, ( r ) ,
w h e r e one f i r s t
0
(by forming
Co = End
NO
(r)
t h e n back up b y a d d i n g a l l o f t h e c o n s t a n t s :
and
MC(r)
would be t o o b i g i n general. ( b ) 4 . 5 2 ( a ) does n e i t h e r h o l d f o r 0 - p r i m i t i v e n e a r - r i n g s with i d e n t i t y nor f o r 2-primitive near-rings without i d e n t i t y ( n o t even f o r
N =
No
and N f i n i t e ) :
129
4d 2-primitive near-rings
r:
n4
N : = {fcMo(T)\f(A)!!A) i s 0 - p r i m i t i v e on r w i t h i d e n t i t y , b u t n o t dense i n M C ( r ) = Mo(r) ( 4 . 2 9 ! ) . M: = { f E M o ( r ) l f ( 3 ) = 0 1 i s If
=
and
A:
= (0,2),
0
2 - p r i m i t i v e on
MC (r)
in
r,
w i t h o u t i d e n t i t y and a g a i n n o t dense
Mo(r).
=
0
(c) A l l 2-primitive near-rinas with i d e n t i t y on w h e r e No ( d ) 4.32,
i s a non-ring,
4.43
and 4.48
Z4,
w i l l b e c l a s s i f i e d i n 4.63.
reduce t h e theorv o f p r i m i t i v e
near-rings t o those o f p r i m i t i v e zero-symmetri c nearr i n g s . We w i l l t h e r e f o r e m a i n l y d e a l w i t h t h o s e o n e s
i n t h e sequel. (e) Recall (4.27(a))
that
Kc
general. H e r e i s some e x a m p l e :
= -x)
r
A!
=
i s "only a set" i n
(f) 0
G = {id,-idl
i s a fixed-point-free
IR.
= {o,id,-id}.
C:
XER:
f(-x)
(with
-id(x):
=
autsmorphism group on
Vc(IR)
= {fEM(IR)If(O)
= 0 A
-f(x)).
n If m c ( R ) = : N , t a k e nl: = sin+-pN and n2: = n C o n s i d e r n : = nlon2EM(IR ) . = id+pN. n no = s i n o ( i d t r ) - s i n ( % ) i s n o t an odd f u n c t i o n , t h u s n o t b e l o n g i n g t o MC(IR), w h e n c e ngN a n d N i s no n e a r - r i ng.
4.54
COROLLARY I f N i s 2 - p r i m i t i v e o n
r
with
AutN 0
(r)
= {id)
t h e n N i s dense i n e i t h e r one of t h e f o l l o w i n g n e a r - r i n o s ( n o t a t i o n as i n 4.52): H o m D ( r , r ) , Maff(r), Mo(r) o r
M(r) 4.55
(cf.
4.65).
THEOREM ( R a m a k o t a i a h non-ring on
r
classes w.r.t.
(2)).L e t
NET),
be a 2 - p r i m i t i v e
w i t h an i d e n t i t y . Then any two e q u i v a l e n c e
2
(except t h e zero class) are equipotent.
$4 PRIMITIVE NEAR-RINGS
130
T*/%
Proof. L e t E be i n
C o n s i d e r t h e map
and
a f i x e d element o f E.
E
f: G
+
E
9
+
q(E)
(with
Since G i s fixed-point-free
(4.52),
aqain).
f i s injective.
so f i s a b i j e c t i o n .
f i s surjective,
By d e f i n i t i o n ,
G=AutN(r)
3 . ) 2 - P R I M I T I V E Z E R O - S Y M M E T R I C N E A R - R I N G S WITH I D E N T I T Y A N D A M I N I M A L LEFT I D E A L . 4 . 5 6 THEOREM ( B e t s c h ( 7 ) , c f . D e s k i n s ( 2 ) ) . w i t h i d e n t i t y which i s 2 - p r i m i t i v e on
r
L e t N = N , b e a nr. and has a m i n i m a l
l e f t i d e a l L . Then %
(a) L = N r' ( b ) 3 e2 = eeL*:
L = Ne = L e
and
morphic t o Proof.
(a) Since
and
+ {ol,
Lr
3
W i t h y as a b o v e , and
m i ni m a l ) . k-%e
Pie
E
i s antiiso-
(eNe;). YEr:
+ {ol,
Ly
so
Ly =
r
Now we c a n a p p l y 3 . 1 0 .
yeel.
= ey
3
(C,,,(T),a)
Hence
(o:y)nL
= Le = L.
eEL":
e y = y.
=
and
e2 = e
{O} a n d L e
By ( a ) ,
Therefore
I01
(since L i s
Le
{Ol.
e2-eEL n ( o : y ) =
SN L,
Since
Le = L .
By L e C N e s L ,
C N ( r ) = CN(L) = CN(Ne)
( i t can
be e a s i l y v e r i f i e d t h a t N - i s o m o r p h i c N-groups have i s o m o r p h i c centralizer-semigroups). For
neN,
consider
d e f i n e d and
t n : Ne xe
+
+
Ne xene
.
i s well-
tn
E C ~ ( N ~ ) .
C o n s i d e r n e x t t h e map
h:
eNe ene
+ +
CN(Ne). tn
If
e n e = erne
so h i s w e l l - d e f i n e d . C l e a r l y , t h e n t n = t,, h i s an a n t i h o r n o m o r o h i s m . I f h ( e n e ) = h ( e m e )
t n = t,
and
Specializing
then
tj X E N : x e n e = t n ( x e ) = t m ( x e ) = xeme. x = :e
we g e t
e n e = eme
and h i s
shown t o b e i n j e c t i v e . Finally,
b
c€CN(IJe)
3
nEN:
c ( e ) = ne.
ene = e c ( e ) = c ( e 2 ) = c ( e ) = ne aet
c ( x e ) = x c ( e ) = xene
tn(xe),
Therefore
a n d f o r a l l X E N we
so c = tn.
4d 2-primitive near-rings
131
and h i s s u r j e c t i v e , hence an a n t i i s o m o r p h i s m . COROLLARY ( B e t s c h ( 7 ) ) .
4.57
N ~ r t ) h a s an i d e n t i t y a n d
If
a minimal l e f t i d e a l L t h e n a l l f a i t h f u l N-groups o f type 2 ( i f those e x i s t ) a r e N-isomorphic
(r,
determines t h e p a i r
I f e i s as i n 4.56(b),
t h e group
CN(r))
u n i q u e l y "up t o i s o m o r p h i s m " .
i s a group w i t h z e r o and L = Ne a s a f i k e d - p o i n t -
(eNe,.) a c t s on
(eNe\{O),*)
( t o L ) and N
f r e e a u t o m o r p h i s m g r o u p (by r i g h t m u l t i p l i c a t i o n ) . Hence e " b r i n g s b a c k " some i n f o r m a t i o n o n 4.58
r
out of NSM(T). C f .
REMARK F o r m o r e i n f o r m a t i o n o n t h e s e t o p i c s ( a p a r t i a l c o n v e r s e o f 4 . 5 6 , t h e u n i q u e n e s s o f (r, CN(r)), s e e B e t s c h ( 6 ) a n d 5 7 a ) , i n p a r t i c u l a r 7.5.
4.)
4.59
COROLLARY L e t 2.50
( f o r "),
(2.50(a)
be a 2 - p r i m i t i v e n e a r - r i n g w i t h
NEW,
5
i s applicable,
hence a l s o
a n d G h a s f i n i t e l y many o r b i t s o n
and 4.21),
i s d i s c r e t e "on
r,
w h i c h i s t h e same a s t o s a y t h a t
MG(r)
Z
(4.29).
7a) f o r the information t h a t i f a f i x e d - p o i n t - f r e e
automorphism group H o f
r
h a s f i n i t e l y many o r b i t s o n
MC(r) h a s t h e DCCL. See a l s o K a a r l i 4.60
etc.)
2 - P R I M I T I V E P!EAR-RINGS WITH I D E N T I T Y A N D MINIMUM CONDITION
DCCL a n d i d e n t i t y . T h e n 4 . 4 6
See
9.227.
r
then
( 2 ) and Oswald ( 1 0 ) .
THEOREM ( B e t s c h ( 7 ) ) . L e t N b e 2 - p r i m i t i v e o n r w i t h D C C f o r t h e l e f t i d e a l s o f No and w i t h a n i d e n t i t y . Then N i s equal t o one o f t h e f o l l o w i n g n e a r - r i n g s ( n o t a t i o n as i n 4.52):
NpNo
N - N o
No a r i n g
Maff(r) H o m D ( r , r )
No a n o n - r i n g
Mc ( r ) 0
MC(r)
dimDr f i n i t e Go h a s f i n i t e l y many
o r b i t s on
r
132
$ 4 PRIMITIVE NEAR-RINGS
P r o o f . f o l l o w s from 4 . 5 2 and 4.59.
Note t h a t
Rc (I') 0
i s a n e a r - r i n g i n t h i s case ( f o r i t equals N ) . 4 . 6 1 COROLLARY I f N h a s a n i d e n t i t y , i s 2 - p r i m i t i v e o n r a n d i f t h e n o n - r i n g No h a s t h e DCCL a n d A u t N ( r ) = ( i d ) 0
N = M(r) ( i f N No) or o t h e r w i s e N = Mo(r). I n b o t h c a s e s , r ( a n d t h e r e f o r e N, t o o ) i s f i n i t e . So t h e D C C i m p l i e s f i n i t e n e s s ! then e j t h e r
4.62
R E M A R K These r e s u l t s i l l u s t r a t e some r e m a r k s i n t h e
preface: w h i l e the "elements o f r i n q theory" a r e r i n g s
o f l i n e a r mappings on r , t h o s e ones f o r n e a r - r i n q t h e o r y a r e n e a r - r i n g s o f a r b i t r a r y mappings ( p e r h a p s w i t h some r e s t r i c t i o n s ) o n
r.
4 . 6 3 THEOREM ( K a a r l i ( 4 ) ) I f
-
then
I a-S z- N
and i f S / I
i s 2-primitive
I d- N .
Proof. Since I i s a 2 - p r i m i t i v e l e f t i d e a l o f S,
I = (L:S)s
h o l d s f o r some 2 - m o d u l a r l e f t i d e a l L o f S b y 4 . 3 . By 3 . 3 4 ,
L 2NS.
Consequently,
Hence
I
=
(L:SINn S
and ( L : S l N 2 N .
I i s an i d e a l o f N .
See a l s o K a a r l i ( 2 ) a n d R a m a k o t a i a h ( 2 ) .
I n t h e l a t t e r paper i t
i s shown t h a t i f N€Hl i s f i n i t e a n d 2 - p r i m i t i v e o n
r,
if N i s a
lrl-1 i s a p r i m e t h e n e i t h e r N=M(T) o r N=Mo(T) n o n - r i n g and i f or ~ N ~ = ~ i r f~ zr ; i s a b e l i a n , N z N rOr h o l d s i n t h e l a s t c a s e ( t h i s r e s u l t can be deduced f r o m 4.55
and 4.61).
133
4d 2-primitive near-rings
5.)
AN APPLICATION TO INTERPOLATION T H E O R Y
4.64
reg
DEFINITION I f
and
NEM(r),
N i s said t o f u l f i l l
the f i n i t e interoolation property I f
w
SE:nU
3
ncN
v
w
Y1#...*YSEr,
Yi
sl:
n(yi)
i c { l #...,
t
Yj
9
for
\I
j
61’..-’6sEr
= 6i.
T h e r e i s an o b t r u s i v e s i m i l a r i t y t o t h e d e n s i t y c o n c e p t s .
In
fact: 4.65
Let
THEOREM
N I M(r)
with
N
and
No
No
not a ring.
Then t h e f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t :
r*,
( a ) No i s 2 - f o l d t r a n s i t i v e o n
r
( b ) N i s 2 - p r i m i t i v e on
Go = { i d ) .
with
( c ) N f u l f i l l s the f i n i t e i n t e r p o l a t i o n property.
Proof.
( a ) ->
(b):
Nr i s t r i v i a l l y f a i t h f u l , 2 - f o l d
t r a n s i t i v i t y implies 1-fold i n turn that Some
6cA”
No6 =
r
and
Nr
{ol.
If
W
ycr
Then A =
r.
t r a n s i t i v i t y and t h i s
I o l 9 A sN r , 0
2
n,ENo:
+
take
no6 = y .
9
So
If Goy 5: G o 6 but y 6 and ( s a y ) 6 o, take 0 . Then noEN 0 w i t h n o y = o A no6 (0:6) , s o y+6 i n (0:Y) NO NO Nor * h e n c e (4.20(c)), a contradiction, Therefore Goy 9 G o 6 some
+
Go = { i d } .
(b)
(c):
b y 4.54
( c ) ->
(a):
trivial.
K a i s e r ( l ) , Lausch ( 5 ) , Ramakotaiah ( 3 ) .
Cf.
and 4.28(d).
P l i t z (12),(13),
P i l z ( 2 5 ) and
134
5 4 PRIMITIVE NEAR-RINGS
4 . 6 6 REMARKS
( a ) So i f a n e a r - r i n g N o f m a p p i n g s o n r i n t e r p o l a t e s a t o and 2 o t h e r p l a c e s then N i n t e r p o l a t e s a l r e a d y on an a r b i t r a r y ( f i n i t e ) n u m b e r o f p l a c e s . Compare t h i s w i t h the corresponding "1 i n e a r " r e s u l t i n r i n g theory ( ( N . J a c o b s o n ) , C o r o l l a r y t o theorem 1 on p . 3 2 ) . This i s a " p u r e l y i n t e r p o l a t i o n - t h e o r e t i c " r e s u l t . ( b ) I t c a n be s h o w n t h a t i f N f u l f i l l s t h e f i n i t e i n t e r p o l a t i o n p r o p e r t y a n d 1i-l > 3 t h e n N o i s a non-
ring. 4 . 6 7 C O R O L L A R Y T a k e r = (IR,+). Then a n y o n e o f t h e f o l l o w i n g n e a r - r i n g s and a l l n e a r - r i n g s c o n t a i n i n q o n e o f t h e m h a v e t h e p r o p e r t i e s t h a t Pi i s 2 - p r i m i t i v e on r w i t h N No, Go = { i d ) and No n o t a r i n g :
+
[XI,N 2 :
t h e n e a r - r i n g o f a l l s t e p f u n c t i o n s on IR the subnear-ring of M(1R) g e n e r a t e d by t h e t r i g o n o metric polynomials.
N1: N3:
= IR
F o r a l l o f them f u l f i l l t h e f i n i t e i n t e r p o l a t i o n p r o p e r t y w h i c h q u a l i f i e s them f o r 4 . 6 5 . The a u t h o r h o p e s t h a t n e a r - r i n g s o f i n t e r p o l a t i n g f u n c t i o n s become i n t e r e s t i n g f o r s p p r o x i m a t i o n t h e o r y ( b e c a u s e t h e s e f u n c t i o n s c a n be i t e r a t e d w . r . t . 0 ) . After a l l t h a t complicated s t u f f the r e a d e r will p o s s i b l y a g r e e w i t h t h e a u t h o r t h a t t h e p r i m i t i v e n e a r - r i n q s have s u c c e s s f u l l y r e v e n g e d t h e i r d i s c r i m i n a t i n g name.
,