A Ternary Interpretation of the Infra-Near Rings

Near-rings and Near-fslds, G.Betsch (editor) 0 Elsevier Science PublishersB.V. (North-Holland), 1987

A

TERNARY

255

INTERPRETATION @F THE

TNFRA-NEAR

RINGS

Mlrela Stefhescu 1.

INTRODUCTION

In studying some g e n e r a l i z a t i o n s of r i n g s a s infra-near rings (see S t e f h e s c u

[lll ),weak

r i n g s (see Climescu [3,41

,

Cupona [5J ) , p r e r i n g s (see Janln [6,7] ),we have noted t h a t cert a i n facts can be b e t t e r explained by means of a ternary i n t e r p r e t a t i o n o f t h e a d d i t i v e composition law. T h i s i n t e r p r e t a t i o n might be of considerable use i n studying a f f i n e infra-near r i n g s ( Stefgnescu LlO3 ) a s well a s i n studying i d e a l s o f those alge-

b r a i c systems. Given t h e development of t h e theory of t e r n a r y groups and t e r n a r y rings,such an i n t e r p r e t a t i o n using a t e r n a r y operation Instead o f t h e binary a d d i t i o n may be i n t e r e s t i n g I n itself. F i r s t we r e c a l l some d e f i n i t i o n s and p r o p e r t i e s o f generali-

z a t i o n s of r i n g s involved I n our considerations. (1.1) Definition. A l e f t h f r a - n e a r r i n g is a triple (I,+,.),

where I i s a nonempty s e t

, + and

(addition and m u l t l p l i c a t i o n )

.

a r e binary operations on I

, such t h a t

t h e following condi-

tions are fulfilled: (1) (I,+) ie a group (generally,noncommutative) i

(il) (I,.)

I s a semigroup;

( i 1 i ) t h e m u l t i p l i c a t i o n is l e f t i n f r a - d i s t r i b u t i v e w i t h r e s p e c t t o t h e a d d i t i o n , 1.e.

x

. (y + z ) = x.y

- x.0

+

X.Z

, for

a l l x,y,z t I .

The concent o f r i g h t infra-near ring is analogous. If x.0 = 0 f o r a l l x ~ 1 , t h e nwe o b t a i n a l e f t near-ring.

If

(I,+,.) i s a l e f t and a r i g h t infra-near ring w i t h a commutative

M. Stefanescu

256

addltion,we obtain an equivalent d e f i n i t i o n o f a weak rhg, t h e general case. We mention t h a t t h e weak rings were first Introduced by A1.Cllmescu 131 in 1961 i n a p a r t i c u l a r case,when x.0

= 0.x = x f o r a l l

X E

I

In 1964,Climescu [41 has noted a ge-

n e r a l possibil.%ltyQf o b t a i n i n g weak r i n g s s t a r t i n g from a g i v e n ring. The p a r t i c u l a r case was rediscovered by G.Cupona

(53 I n

1971, who c a l l e d it "quasi-ring".

Let ( I , + , . ) be a l e f t infra-near ring. One can e a s i l y check that f o r each

- x.0

, is

X G I t h e mapping f x r

I--+I,given by fx(y)

an endomorphism of the a d d i t i v e group (I,+)

.

I=

-

x.y

This a l -

most obvious remark may help us t o study t h e l e f t infra-distribut i v e a s s o c i a t i v e multiplications over an a d d i t i v e group (see Stefgnescu (123 ) , t h e discussion being unexpectedly i n t e r e s t i n g .

One of t h e first examples o f r i g h t near-rings was the s e t of a l l a f f i n e transformations over a vector space or over an Abelian a d d i t i v e group endowed w i t h the pointwise addition and the mapping composition. This is a l s o a l e f t infra-near ring and a

few p r o p e r t i e s of t h i s s e t come from i t s infra-near r i n g struc-

ture

. It is s u r p r i s i n g how much algebra can be obtained only by

considering t h i s weaker s t r u c t u r e (see ClO'J

,a

?aper i n which we

have studied and generalized the infra-near ring of a f f l n e trans-

formations from t h i s point o f view ). If a l e f t infra-near ring s a t i s f i e s the condition I 0.x = 0

, for

all xcI,

then we c a l l it a l e f t C-infra-near 0.x = x

, for

all x

rinp;

, while

in the case

I

61,

we c a l l it a l e f t Z-infra-near

ring.

we give two examples o f f i n i t e l e f t C-infra-near rings: .'1

The c y c l i c group (&,,+),with

1.x = 2.x = x , f o r a l l x e Z 3

.

t h e multiplication

I 0.x

= 0,

A ternary interpretation of the infra-near rings

.'2

The symmetric group S3

3:

where 3a = 2x = 0 and x+a = b

257

{O,a,b = 2a,x,y = x+a,z

+

, with

x

x.8

= y.s =

z.8

= a

, b.8

=

(11) 0.8 = 0,a.s = x . 6 = y.6 = a, b.s = z . s =

(111) 0.8 = 0,a.s = x . 8 = a , b.8

P

y.6

P

x+b],

t h e multiplicatlon given

by one of the following t h re e p o e s l b i l l t i e s r

(I) 0.8 = 0,a.s =

a

z.8 =

B

, for a l l

BC

%'

8

,for a l l s e S 3 1

8

,for a l l B E + .

Many o t h e r eaamples of l e f t infra-near rings and a general s t u d y of such algebraic systems can be found in [llJ

mark that

, If

. Let u s re-

(I,+) is a group,then (I,+,+)I s a l e f t and a

right

lnfra-near ring. I n a l e f t lnfra-near r i n g we do not obtain 0.0 = 0. For examp l e , i f we consider I = 25 and the binary operations I x

+ y = (x1+91,72+92,X?)+Y3

,XI++Yl+,x5+x2 .Y3+Xl+.Yl+Y51#

x.y = ( X I .Y1,'2*92,a~ ,a&, w i t h a3,a4,a5 fi x e d elements in Z,then we obtain a l e f t infra-near ring for which

0.0 = (0,0,a3,a4,a5),whlle 0 = (0,0,0,0,0).

The r e a l difference between t h e general case (see L83

in

which th e multiplication is an "independenttt operation on an ad-

d i t i v e group,

- that means the multlplication

cond!tions w i t h respect t o t h e addition,

does not f u l f i l any

- and our case

( I n which

t h e multiplication I s l e f t l n fra -d i st r i b u t i v e over t h e addition) is perceptibly I n the theory of congruences,hence i n defining homomorphisms and Ideals of such st ru ct u res . (1.2) Definition. A fno-Slded Ideal of a l e f t infra-near ring (I,+,.)I s a nonempty subset J o f I s a t l s f y l n g the conditions: (I(J,+) ) is a normal subgroup of t h e group

(ii) x . j (ill ) ( j

- x.0 e J

+

x).y

, for

a l l x E I and j E J

- x.yeJ , f o r

(I,+)i

i

a l l x , y E I and

j C J

.

If J o n l y satisfies t h e conditions (Iand ) (ii) (reSpeCtiVdg, (Iand ) (ill )) , J I s c a l l e d a l e f t Ideal (respectlvely,a right

I d e al) of

I.

M. StefGnescu

258

If (I,+,.)is a l e f t near-ring,hence x.0 = 0 for a l l x e 1 , t h e n

we obtain t h e d e f i n i t i o n s o f the two-sided,left and r i g h t idealB of a l e f t near-ring.

The similitude is a m a s i n g , B u t i n f a c t the

s i t u a t i o n 1s more complicated

, since

even t h e two sided i d e a l s

of a l e f t infra-near ring a r e not closed under the multiplloa-

tion,hence generally they a r e not subinfra-near r i n g s o f I. For l e f t near-rings a s i m i l a r assertion is t r u e only f o r the r i g h t ideal8 (see P i l z L9l

, 1.28

and 1.33).

O f c o u r s e , i f J is a two-sided i d e a l of the l e f t Fnfra-near

ring I

, then

I/J = f x + J / x

E

I) i s a l e f t infra-near r i n g with

respect t o t h e usilal operations beween cosets.

(1.3) Definition. Let (I,+,.)and (It,+,.) be t w o l e f t infra-near r i n g s . A mapping y t 141’ i s c a l l e d a homomorphism of l e f t infra-near rings i f t h e following two conditions a r e f u l filled : (1)

‘P(x + Y) = \ p ( x ) + \ P ( Y )

,

.

fix). y(y) , f o r a l l x , y c I , k e r ‘4 = jx E I / yJ (x) = 0’5 , i s a two-sided

(ii) .p(x.y) =

The kernel of ‘f

i d e a l of 1,while t h e image o f p

, Im y

=

(p(x> / x E I

a subinfra-near ring of 1’. Moreover,the %-sided

f , is

i d e a l s of I

a r e exactly t h e kernels of the homomorphisms from I t o another l e f t infra-near ring

.

(1.4) Definition. Let I be a l e f t C-bfra-near be a group. If t h e r e e x i s t s a mapping

= m.x

, such

(I) m.(x

MxI4M

that t h e following axioms a r e f u l f i l l e d

+ y) = m.x

(ii) (m.x).y

= m.(xy)

- m.0

+ m.y

,p ( m , x )

(M,+,p) a right

, f o r a l l m h M and x,y 0 I 4

, I-group.

Obviously I is i t s e l f a r i g h t I-group and each right i d e a l of I i s a r i g h t I-subgroup

, while

=

I

, f o r a l l m cM and x,ye I ;

(111) 0.x = C , f o r a l l x C I

then we c a l l

,441

r i n g and (M,+)

a r i g h t I-subgroup of I i s

259

A ternary interpretation of the infra-near rings

n o t a r i g h t ideal of 1,even in th6 cam when it i e a normal oubgroup of (Il+).

Denote by D = i d G I / d.0 = 0 5 and W

wrI / w.0 =

o

w),

t h e s u b s e t of a l l l e f t d i s t r i b u t i v e elements of I and t h e s u b s e t of

a l l weakly l e f t d i s t r i b u t i v e elements of I , r e s p e c t i v e l y . I f I

i s a l e f t C-infra-near

x,yr:I

, hence

+

ring and (x

y).O

e

x.0

+

y.0 f o r

0 i s a r i g h t d i s t r i b u t i v e element of I

all

, then

D is

a l e f t subinfra-near r i n g of I which i s a normal eubgroup of (Il+)

.

and W i s a r i g h t I-subgroup of I

r e c t decomposition I = D + W x = (x

- x.0)

2. TERNARY

+

, where

x.0

x

;

Moreover w e have t h e semidi-

indeed,for any x E I ,we have

- x.OED

.

and x . O E W

GROWS

Let u s note t h a t t h e l e f t i n f r a - d i s t r i b u t i v i t y of t h e m u l t i p l i c a t i o n w i t h r e s p e c t t o t h e a d d i t i o n reminds one of a l e f t distri-

b u t i v i t y of the binary multiplication over a ternary composition defined by

8

( y , t , z ) := y

-t+z

(y,@,z) = y

Indeed,we have then for a l l x l y l z 6 1

.

for all x , y , t , z 6 1

for a l l y , t l z GI.

+

Moreover we have x.(y

.

= (x.y1x.Olx.~)

z and x.(y,O,z)

-t

+ z)

= xdy-x.t+x.z,

B u t such t r i p l e s were obtained by Dijrnte

and by Baer i n t h e t w e n t i e s i n connection w i t h some g e o m e t r i c a l f a c t s

( t h e Erlangen Program on groups) ; In

[11

,R.Baer had n i c e l y ex-

plained the geometrical and algebraic reasons to take a s Fnva-

riants f o r a group of transformations t h e expressions of t h e form x

.

-y+z

S t a r t i n g f r o m t h i s point

Certaine

was

guided t o consider a ternary operation on a group (G,+) defined by

(x,y,z)

I=

x

-y+z

f o r a l l x,y,z E G (we t r a n s l a t e i n t o an

a d d i t i v e notation t h e paper written by Certaine)

.

By considering

t h e a b s t r a c t properties of such an o p e r a t i o n , C e r t a h e has defined ternary groups.

M.Stefdnescu

260

(2.1) Definition. Let I! be a nonemptg eet endowed with a tar-

nary composition

,.) : T

(.?.

%

T x T 4 T and With a f i x e d element

O d T such that t h e following conditions a r e f u l f i l l e d : (1) (x,O,(y,Z,u)).= ((x,O,y),z,u)

, for a l l x t T , for a l l x e T

(2) (x,O,O) = x

(3)

(x,x,O) = 0

, for

a l l x,y,z,u e T ;

.

;

Such an alg ebra i c system i s called a ternary group of first type

(Certaine 121

, Definition

3

By defining a binary operation on T

(4) x

+ y r = (x,O,y)

, for a l l

x,yeT

,

we obtain a group (T,+) w i t h 0 a s i t s n eu t ral element ( 123

Theorem 4 )

.

,

?%en t h e binary and t h e ternary operations a r e r e l a t e d by t h e

eq u ali ty :

(5)

(x,y,z) = x

- y + z , f o r a l l x,y,z

E T

,

then we say t h a t t h e ternary group i s reg u l ar

, the

In a binary group

n e u t ra l element is unique

statement does n o t hold f o r a ternary group satisfy (11,

( 2 ) and ( 3 ) i n D e f i n i t i o n ( 2 . 1 1 ,

, but

.

A s i mi l ar

if 0 and O1 both

then t h e cor-

responding binary groups by (4) a r e isomorphic a s it i s proved in

(21

( t h e isomorphism i s given by x ~ ( x , O , O 1 )

, for

xe!&

Moreover it i s possible t h a t d i s t i n c t ternary groups correspond t o the same binary group,aa it is shown

in c2’J f o r t h e

c y c li c group w i t h two elements. To i l l u s t r a t e this,we consider th e cy c li c group o f order 3 3a = 0

,

(T = {O,a,b) ,+) w i t h b = 2a and

. We f i n d three p o s s i b i l i t i e s t o define t ern ary group8

o f f i r s t type,namely

t

(I) the r egu l a r ternary gmup (x,y,z) = x X,Y,Z

-y+

6T 3

(11) the s p e c i a l products a r e (a , a , a ) = (O,b,b) = b

, f o r my

, (b,b,b)

=I

26 1

A ternary interpretation of the infra-near rings P

(O,a,a)

a

L

, (b,a,a)

(a,b,b)

o

, the

0

other ternary pro-

ducts being r e g u l a r ;

(a,a,b) = (b,b,a) = 0

(111) the s p e c i a l products a r e

, (O,a,b)

(O,b,a) = (b,a,b) = a

= (a,b,a) = b

, the

,

other pro-

ducts being regular, ~e obtain anotkz concept of ternary group, which we call

of

seccpld

type,

type, while a ternary group of Certain's type is called of f & s t type.

(2.2) Definition. A couple ( T , ( . , * , . ) )

.,

set and (., .)

I

Tx Tx T

,where T i s a nonempty

, if

called a t e r n a r y group of second type

it s a t i s f i e s the

following axioms f o r a l l x , y , z E T and a fixed element O € f ( 6 ) ((X,Y,O),Z,O)

(7)

(O,X,O)

(8)

((O,O,x),x,o>

= x

,

=

(X,(YtZ,O>,O~

, then

(T,+)

, where

defined by (9)

+

x

y

I=

I

,

= 0

(2.3) Proposition. If ( T , ( . , . , . ) ) cond type

, is

i s a t e r n a r y operation on T

3T

(x,y,O)

i s a t e r n a r y group o f set

, for

i s t h e binary composition

a l l x,yC:T

is a group.

Proof. The a s s o c i a t i v i t y

comes from (6)

, , by

using the defi-

n i t i o n o f t h e binary operation. The other two axioms t r a n s l a t e d into the addition complete the definition of a group. We a l s o have the equalities:

,for

(7')

(x,O,O)

(8')

(x,(O,C?,x),C) = 0

= x

all x%T

, for

,

all xbT ,

(2.4) Definition. A t e r n a r y group of second type i s c a l l e d

regular

,

i f the ternary operation is connected w i t h t h e binary

composition (9) by the following equality!

(lo)

(x,y,z) = x

+y

-z

, for

all x , y , z E T

.

As in the case of t e r n a r y groups of first type, t h e r e

are

M. Stefanescu

262

many t e r n a r y groups of second t y p e a s s o c i a t e d t o t h e same group by t h e e q u a l i t y ( 9 ) . For example, for t h e c y c l i c group of o r d e r 2 , (T = { O , a ) , + ) , we may d e f i n e a t e r n a r y composition: ( O r O r O ) = = (a,O,a) = (a,a,O) = ( a , a , a ) = 0, (O,a,O) = (O,O,a) = ( a t O , O )

=

= ( O , a , a ) = a , which p r o v i d e s T w i t h t h e s t r u c t u r e of a t e r n a r y g r o u p of second t y p e w i t h (T,+) a s t h e a s s o c i a t e d group. T h i s t e r n a r y group is n o t r e g u l a r , s i n c e ( a , a , a ) = 0 # a = a + a a.

-

The following ternary compositions on T = \O,a)

,namely (7), (8)

only two of the three axioms I n Definition (2.2) for (tl),

(61,031 for (t,) and (6),(7) for (t,)

I

{

o

(O,O,O) = (a,O,O> = (O,a,a) = (a,a,o) =

satisfg

,

(O,o,a) = (a,O,a) = (O,a,O) = ( a , a , a ) = a ; (0,0,0)= ( O , O , a )

= (o,a,o) = (O,a,a) = 0

,

= (a,O,a> = (a,a,O) = ( a , a , a ) = a ;

lo ,o,o)

= (a,O,a) = (O,a,a) = 0

(t3) (O,O,a) = (a,O,O) = (o,a,o)

b

= (a,a,O) = ( a , a , a ) = a

.

These uodels prove t h e following

(2.5) Proposition. The system of axioms in Definition (2.2) is independent. (2.6) Definiti0n.A ternary group

(of first or of

(I,(.,.,.))

second type) together w i t h an a s s o c i a t i v e binary m u l t i p l i c a t i o n which i s l e f t d i s t r i b u t i v e w i t h respect t o the t e r n a r y composi-

t i o n , i . e . which s a t i s f i e s the e q u a l i t y (11) x.(y,z,w)

= (x.~,x.z,x.w)

, for

a l l x , y , z , w ~ I,

i s c a l l e d a l e f t t e r n a r y near-ring (of first o r of second type).

(2.7) Fropositlon.(l) If ( I , ( . , . , . ) , . ) is a l e f t t e r n a r y near-ring of first type

, then

(I,+,*), where + is defined by (4),

.

., ,

is a l e f t infra-near r i n g , (ii)If (I,(., .)

i s a repular

l e f t ternary near-ring of second type, t h e n (Il+, defined by (9)

, is

a l e f t infra-near ring.

,where + is

263

A ternary interpretation o f the infra-near rings

Proof. (i) Since (I,+) I s a group (Certalne 127, Theorem 4) , and (I,.) is a semlgroup by assumption, we have to check the left

i n f r a - d i s t r l b u t i v i t y of t h e m u l t i p l i c a t l m w i t h r e s p e c t t o t h e addition. We have

I

= ((x.y,0,0>,x.0,x.z) But x.0

+

(O,X.O,x.Z)

= (x.O,x.O,x.z) - - x.0 + X . Z y,zGI

.

,

+ z ) = x.(y,O,z) = (x.y,x.O,x.z)

x.(y

= (x.y,0,(0,x.0,x.z~) =

+

, therefore

X.Z

z ) = x.y

(ii)Here we obtain immediately

= (x.y,x.z,x.O)

= x.y

= x.y + (O,X.O,X.Z).

= (x.O,O, (O,X.O,X.Z))

= x.(O,O,z) hence x.(y

+

- x.0

X.Z

x.(y

-

=

= ((X.O,O,O) ,x.O,X.Z)

(O,x.O,x.z) f x.0 + xlz , f o r all x,

t z)

, from

= x.(y,z,O)

=

t h e r e g u l a r i t y of t h e

t e r n a r y group. B u t taking y = 0 we obtain (12)

therefore

x.z

+

x.(y

x.C = x.0

+ z) = x.y

+

X.Z

- x.0

,for a l l x , z e I

+

X.Z

The o t h e r exioms i n the d e f i n i t i o n o f

,

.

, for

a l l x,y,z € 1

left

infra-near r i n g s

come from t h e hypotheses and t h e Proposition (2.3)

.

Ye note t h a t i n the g e n e r a l case t h i s c o m u t a t i v i t y g h e n

, as

by (12) does n o t h o l d f o r l e f t infra-near rings

one can

see from the exemple I = E5 i n Section 1.

T h i s way we obtain a s a p a r t i c u l a r case of t h e t e r n a r y near-

-rings the ttpreringsttdefined by Janin [6,73

.

Indeed, a p r e r h g

is a ternary near-ring of first type or of second type which satisfies for all x,y,zqI a commutativity condition of the ternary com-

position

, namely

(x,y,z) =(y,x,z)

(x,y,z) = (z,y,x)

- for

- f o r the

first type and

t h e sec-nd e T e ,which has a n idempotent

m u l t i p l i c a t i o n being l e f t and r i g h t d i s t r i b u t i v e w i t h r e s p e c t t o t h e ternary composition.

We t r a n s l a t e the d e f i n i t i o n of a two-sided i d e a l of a l e f t infra-near ring into ternary language. In a left ternary near-ring,

we say t h a t a nonempty subset J i s a n idea1,lf it t h e following conditionsr

satisfies

M.Stefdnescu

264

, ((x,O,a),x,O)e

J

(i) (a,b,O)(

J

for a l l a , b € J and ~ € 1 1

, for a l l a d J and x G I ; (ill)((x,O,a).y,x.y,O) € J , f o r all a E J and x , y e I (ii) (x.a,x.O,O)

E J

.

Let I be a l e f t ternary near-ring of first type i n which x.0 f 0 b u t 0.x = 0 f o r a l l x G 1 . In addition, we assume t h a t f o r any x,y( I

, (x,O,y).O

, hence

= (x.O,O,y.O)

i s r i g h t d i s t r i b u t i v e with respect t o the

t h e element 0

t r i p l e s o f t h e given

, a s defined above, which a r e semigroups under multiplj.catlon , a r e a l s o closed under t h e t e r n a r y operation. We note t h a t x.wEW , for any X E Iand w e W

form. Then t h e subsets D and W

In f a c t , t h e following proposition holdsi

Proposition. Let I be a l e f t ternary near-ring of f i r s t

(2.8)

type which s a t i s f i e s t h e conditions

= x

(I) 0.x

, for

(ii)(x,y,z).O

I

a l l XO ; I

, for

= (x.O,y.O,z.O)

a l l x,y,zeI ;

, for a l l X Q ; I (w.x,O,d.x) , f o r a l l d c D , w e W and

(iii) (O,x.O,x) = (x,x.O,O)

( i v ) (w,O,d).x =

/ deD,weW

Then : (i) I = (D,G,W) = i ( d , O , w ) x

f

xaI.

and foa? each

e I , t h e elements d t D and w c g W such that x = (d,O,w)

a r e uni-

quely determined; (ii)Using ternary expressions f o r two a r b i t r a r y elements

of I

,

the multiplication is given by t h e formula8

= (d.d',O,(w,O,d.w'))

(13) ( d , O , w ) . ( d ' , O , w ' >

Proof. (i) For each -

x C I

belongs t o D and w = x.0 EW

, we

.

determine d = (x,x.O,O) which w i t h d and w j u s t

B u t x=(d,O,w),

determined above (one can verify this relation by straightforward calculations). w,wl t w

, we

have

If (d,O,w)

= (d',O,w')

, with

d , d l E D and

=

(((O,d',O),O,(d,O,wR)),O,(O,w,O))

= (((0,d',0~,0,(d',0,w~),0,~0,w,0~~ and after some skillful calcu-

l a t i o n s we obtain = (w',w,O)

(0,d:d) = ( w ' , w , O ) C W n D

= 0 , B u t then d1 = ( d ' , O , O )

, hence

(O,d',d)=

= (dl,O,(O,dl,d)) =

265

A ternary interpretation of the infra-near rings

= ((d',OiO),d',d)

(d',d',d)

p

(d',d',(O,O,d))

p

= (O,O,d) = d

t((d',d',O),O,d)

0

I n t h e same manner, we prove

that w' = w

(it) The formula (13) is obtained by using t h e condition (iv) the e q u a l i t y w.x = w f o r a l l x 6 1 and w t w .

t a k i n g i n t o account

It might be Fnteresting t o study ternary near-rings In an Independent context

. Here we

have used t h e t e r n a r y i n t e r p r e t a -

t i o n f o r showing t h e r e l a t i o n between infra-near rings and pre-

rings and f o r explaining t h e l e f t l n f r a - d l s t r i b u t i v i t y a s a ref l e c t i o n o f a l e f t d i s t r i b u t i v i t y of t h e m u l t i p l i c a t i o n with r e s p e c t t o a t e r n a r g composiflon. REFEFf ENCES

C11. Baer,R.

- Z u r EinfUhrung des Scharbegriffs, J.reine

Math. ,160 (1929) C23. Certaine,J.-

, 199-207.

The t e r n a r g operation (abc) = ab-lc of a

group, Bull.Amer.Math.Soc.,49

c3]. Climescu,Al.-

Anneaux f a i b l e s

(11) (1961)

ClFmescu,Al.

[4],

9

(1943)

, 869-877.

, Bul.Inst.Polit.Ia$i

,7

1-6

- A new c l a s s of weak rings , (Romanian) ,

ibidem, 1 0 ( 1 4 ) ( 1 9 6 4 ) , 1 - 4 .

[5]. Cupona,G.

angew.

- On

quasirings

Phys,ldaC$doine

, (Macedonian

, 20

, Eull.Soc.lyfath.

)

(1969) ,19-22 (1971) 4 MR

44 #I703

c63. J a n h , P . - Une g h 6 r a l l s a t i o n de l a notion d'anneau.Pr6-

anneaux.C.R.Acad.Scl.Paris,Sec.A

, 269

(1969)

,

62-64

[7].

Janln,P.

- Une g h 6 r a l i s a t i o n de l a notion

Prealghbres. C.R.Acad.Sci,mris

(81

.

(1969)

, 120-122.

Murdoch,D.C. ,Ore,O.

d'algbbre.

, Sec.A , 269

- On generalized rings, Amer. J.Math. ,

M.StejZnescu

266

63 (1941) 973-78

0

[91. P i l e ,G.- Near-rings .The theory and its applications

North-Holland Mathematics Studies 23

, North-Holland

Pub1 .Comp. ,Amsterdam, 1977.

[lo].

Stefhescu,MFrela

- Infra-near

An.St.Univ.Al.I.Cuza

1111.

Stefhescu,Mirela

-A

r i n g s o f a f f i n e type,

I a s i , 2 4 (1978) ,5-14

generalization o f the concept of

n e a r r i n g rInfra-near r i n g s

45 [12].

- 56

9

StefZinescu ,Mirela

,

ibidem, 25 ( 1 9 7 9 ) ,

- Multiplications i n f r a - d i s t r i b u t i v e 6

s u r un groupe,Publ .hth.Debrecen, 255

- 262.

.

27 (198C3,