Chapter 7 The Theory of Generalized Identities

CHAPTER 7

T H E T H E O R Y OF GENERALIZED IDENTITIES In this chapter we build a theory of generalized identities (GIs) from the foundation laid in Chapter 2. Not only is this theory very pretty and natural, arising merely by extending the set of coefficients, but it provides a very useful way to encode specific information about rings and their elements. Several important PI-theorems turn out to be consequences of more general, easy-to-prove facts in the GI-theory ; Bergman’s clear proof of the first fundamental theorem on rational identities (48.2). as well as several applications, are also based on GI-theory. The heart of GI-theory lies in the structure of primitive rings. Accordingly, the “socle” of a ring is defined and described in $7.1 in sufficient generality to be applied later to the (*)-primitive case. as well as the primitive case. $7.2 contains most of the major structure theorems of GI-theory, stemming from a technical result describing the “evaluations” of a generalized monomial on a primitive ring. In 47.3 we look at an involutory analogue of “primitive,” leading to the extension of the structure theorems to the (*)-case in 87.4. One immediate application is a famous theorem of Amitsur, that (*)-PI implies PI. In 47.5 we bring in an important ultraproduct technique from logic to yield decisive results on prime and (*)prime rings; these results are tightened in $7.6 by using Martindale’s “central closure.” Various applications to PI-theory are given in the text as well as in the exercises. 47.1. Semiprime Rings with Socle

In the general study of primitive rings, and more generally of semiprime rings, the notion of “socle” arises naturally in several contexts. In this section we examine some basic properties of the socle with the specific intention of applying them to primitive and (*)-primitive rings, in $7.2 and 47.4, respectively. Intuitively, we want to see how closely primitive rings with minimal nonzero left ideals resemble matrix rings over division rings. Throughout 1hi.Jsection, R denotes a semiprime ring. Both here and in (j7.3 we shall carry over standard methods (cf. Herstein [76B, Chapter 1.21); for 254

[g7.1.]

Semiprime Rings with Socle

255

more insight into rings with socle, the reader should consult Jacobson [64B, Section 41. Definition 7.1.I. The socle of R, written soc(R), is 0 unless R has a minimal (nonzero) left ideal, in which case soc(R) = x(minima1 left ideals of R ).

First we suppose that soc(R) # 0, i.e., R has a minimal left ideal L. Note that for any nonzero idempotent e of L, we must have x = .ye for all x in L. (Indeed, L = Re, so x = re for some r in R, implying x = re = re2 = xe.) Proposition 7.1.2. Suppose U E L and La # 0. Then La = L and = 0, and L has an idempotent e such that ea = a = ae.

Ann,a

Proof. La and Ann,a are left ideals of R contained in L ; since L is minimal, we get La = L and Ann,a = 0. Thus a € La, so a = ea for some e E L. Then ea = e2a, so (e2- e ) E Ann,a = 0, proving e is idempotent ; hence a = ae. QED Corollary 7.1.3. L has a n idempotent.

Proof.

L2 # 0 since R is semiprime, so La # 0 for some U E L. QED

Of course, L is an irreducible R-module. Thus End,L is a division ring, by Schur’s lemma. Note that for any idempotent e of R, eRe is a ring (with multiplicative unit e ) . Proposition 7.1.4. For any idempotent e of R, End,Re = eRe, under which the right module operations of Re on End, Re and eRe are ident$ed.

Proof. Define $: End,Re -+ eRe by $(p) = ep(e)EeRe for p in End,Re. Then $(p) = p ( e 2 )= B(e), so $ is clearly a ring homomorphism, and ker $ = {BE End,Relp(e) = 0} = 0. Given r in R, define p, in End,Re by right multiplication by re, i.e., f i r ( r ’ e )= r’ere. Then $(p,) = ere, so $ is an isomorphism. Also, for each re in Re and /3 in End,Re, p(re) = rep(e) = re$@), identifying the module operations. QED Corollary 7.1.5. I f e is an idempotent of L, then eRe is a division ring. Proposition 7.1.6. Rr is a minimal left ideal iff rR is a minimal right ideal.

Proof. (*) Suppose 0 # x ~ r R It. suffices to prove r E x R . Rr has an idempotent e, so r = re. Write x = ra = rea. Then reaRrea # 0, so for some r‘ in R , 0 # rear’reExR. But eRe is a division ring, so ear’re is invertible (with respect to e ) , implying re E xR. So r = re E xR. (c) By left-right symmetry. QED

256

rHE THEORY OF GENERALIZED IDENTITIES

[Ch. 7

Corollary 7.1.7. soc(R) = 0 unless R has minimal right ideuls, in which case soc(R) = x(minima1 right ideals of R). Hence soc(R)a R.

Proof. If r # 0 is in a minimal right ideal (of R), then rR is a minimal right ideal, so Rr is a minimal left ideal, implying rEsoc(R). The rest is immediate. QED Proposition 7.1.8. I f R is prime, then for any 0 # AQR we have

soc(R) s A.

Proof. This is trivial unless soc(R) # 0. Then for every minimal left ideal L we have 0 # AL c L, so L = AL E A. Thus soc(R) = z(minima1 left ideals) c A. QED

We come to another way of looking at the socle. For the remainder ofthis section, assume M is a faithful, irreducible R-module, and let D = End, M, a division ring. For each r in R, rM is a D-subspace of M, so we can define M-runk(r) = [rM:D]. When there is no ambiguity about M, we write rank for M-runk. Remark 7.1.9. For all r,, rz in R rank(r,r,)

< rank(r,), and rank(r, + r 2 ) < rank(r,)+rank(r,).

rank}

a R.

< rank(r,), rank(r,r,) Thus {elements of finite

Our main objective is to identify soc(R) with the elements of finite rank. Remark 7.1.lo. Suppose V is a D-subspace of M . If rank(x) = m and y1,..., y, are elements of V such that xy, ,..., xym are Dindependent, then we can expand y1,...,y, to a basis (y,,y, ,... } of V

-= cc

having the property x y i = 0 for all i > m. [Indeed, by Zorn's lemma, we can find a maximal D-independent set T = {y,,y,, ...} of V having this property. For any y in I/, we have x y E X:' , x y , D ; writing xy = ,xyidi, we have ~ ( y - Z ~ ! ~ y =~ 0, d , )so V - ~ ~ = , y i disi spanned by T, by maximality of T, implying y is spanned by T.]

xy=

In the above remark, the basis of V need not be countable although, merely for convenience, the notation {y,, y,, .. .} suggests that the set is well ordered. Lemma 7.1.11. If rank(r) = 1, then Rr is a minimal leji ideal of R. Proof. Rr # 0 since r # 0, so it suffices to prove for every x # 0 in Rr that Rr E Rx. Suppose ry, = y', # 0. Let T = {y,, y,, . . .} be a D-basis of M such that ryi = 0 for all i 2 2. Writing x = a,r for suitable a, in R, we have a,y; # 0 (or else x.44 = 0, contrary to x # 0). Take rl such that rIu,y', = y,. Then (rlulr)yi= rj.; for all y i in T, implying (r1a,r--r)yi= 0, so r = r l a , r = r,xERx. QED

$7.2.1

The Basic Theorem

257

Lemma 7.1.12. ff r e R arid rank(r) = t 2 1, then there are rank 1 elements r l , . . . ,r, in Rr such that r = Xi= Iri.

xiz

y,D for suitable D-independent y i in M , and Proof. Write rM = pick xi in R such that x i y i = yi and xiyj = 0 for all j # i. Then for any y in M we get suitable di in D such that ry = & l y j d j = C:,j=lxiyjdj xi(ry) = I:=( x i r ) y ;letting ri = xir, 1 d i 6 t , we have r-Z:= ri = =O. QED Theorem 7.1.I 3.

soc(R) = {elements offinite rank}

Proof. ( 2 ) If r E R and rank(r)=t, then by Lemma 7.1.12 Rr = Rri for suitable rank 1 elements ri of R ; hence by Lemma 7.1.1 1 Rr is a sum o f t minimal left ideals. Hence r E soc(R). ( c )We are done unless soc(R) # 0. Then by Proposition 1.5.9 we may assume M is a minimal left ideal L of R. By Remark 7.1.9 and Proposition 7.1.8 we need only find an element of finite rank. Let e be an idempotent of L. Identifying eRe with End, Re as in Proposition 7.1.4, thereby obtaining an injection R --t End(Re),,, (given by left multiplication), we have rank(e) = 1, as desired. OED

x.f=

Our treatment of primitive rings relies heavily on Theorem 7.1.13; our main strategy will be to work the socle through manipulation of elements.

87.2. The Basic Theorem of Generalized Polynomials and Its Consequences

In $1.5 we commenced the study of PI-algebras by proving Kaplansky’s theorem in two parts; (i) by the density theorem, and the staircase, each primitive PI-ring has the form M , ( D ) ; (ii) by closure, [D:Z(D)] < co. Replacing PI by GI (generalized identity), we shall obtain Amitsur’s theorem [65a], in which conclusion (i) is weakened to the following: (i’) The socle of a dense subring of EndM, satisfying a proper GI is nonzero; conclusion (ii) remains intact, that is, [ D : Z ( D ) ]< co. [There is an interesting monomial condition that is weaker than “GI,” given in Appendix B, equivalent to (i’).] Actually, a more explicit result will be given, that every GI of a primitive ring can be rewritten in such a way that each monomial contains a coefficient of bounded rank. This theorem is very powerful, even yielding nice applications to the PI-theory. We start by assuming R is a primitive ring with faithful, irreducible module M . Our basic goal is to find some notion to take the place of a

258

[Ch. 7

THE THEORY OF GENERALIZED IDENTITIES

“staircase” of matric units. More specifically, writing a GI of R in the form il sS>m(t) i,

=1

we want to find x , , . . ., st in R and y in M such that rinlxn,. ‘ . . ~ , ~ r l~y, = , ~ 0+ unless 71 = I and i = 1, and r l ,.yl . - - x f r l , y # 0. Finding such elements is extremely complicated, although one can simplify matters by reducing to the case t = 2 (cf. Exercise 1). We approach this idea inductively. working only on the rinr. We shall carry some notation through Theorem 7.2.2. D = End,M. Letting F be a maximal subfield of D, form R F as in Proposition 1.5.12; then M is a faithful, irreducible RF-module, with F End,,M. We consider a multilinear generalized polynomial f ( X I , . . ,X , ) of ( R F ) { X ) . Viewing M as F-vector space, let “subspace” denote finite-dimensional F-subspace of M . Given subspaces V,, . . . , y , we say ,f is (V,, . . . , I/;)-oalued [or just (I/)-ualued]if for all ri in RF such that ri = 0, 1 -< i < t , we have , f ( r l , ..., r,)M s V,. Remark 7.2.1 .

Iffl, J; are (I/;)-valued, then (fl

+.f2)

is (v)-valued

Theorem 7.2.2 (Notation as above) S u p p o s e , f ( X 1 ,...,X I ) is u sum of’ u monomials und is (I/,)-ualuedf o r suitable subspaces V,, . . . , V,; let u = max([v: F]IO < i < t ) . Let W, be the F-subspace q f R F spanned by the coeficients of J: 7hen f can be written as a sum of
coeficients in W,, each monomial hauing a coeficient of rank over F .

611

+iu(v-

1)

Proof. Call a generalized polynomial good if it can be written as a sum of monomials (with coefficients in W,), each having a coefficient of rank ,< u +$v(v- 1). We shall show f is good by using simultaneous induction on t and u. First note that if t = 0, then .f is a constant w ; by hypothesis, wM E V,, so w has rank < [V,:D] = u and we are done. Next, for any t if u = 0, then .f’= 0 and we are done. Thus we may assume t 2 1 and u 2 1. Write

f=

1

C C hij(X1,.. . , X i - 1, X i + 1 , . . .,Xr)Xiwij, ,= j = I 0 4

1

xi=

where each hij is a monomial [in ( R F ) { X } ] thus ; ui = u. For each i, we reorder the hij if necessary such that for some v; < ui we have w,M c for all j such that uf < j < ui. with u; chosen minimally (possibly with (L = 0). The reason for considering u; is twofold. First, for all j > u:, wij has rank <[I/:D] < u ; secondly, for all j > u:. if ri I/ = 0, then riwijM = 0, proving h i j X i w i jis (0, V,, . . , K)-valued.

<

$7.2.1

259

The Basic Theorem

Let f ’ = Cj=lZyL hijXiwijand f ” = f - f ’ . Now f ” is good, as well as (0, Vt,. . ., c)-valued. Hence, we are done i f f ” = f . Even iff” # 0, we see that ,f’ is (V,, V,, . . .,T/;)-valuedby Remark 7.2.1, and thus by induction on u is good ; hence f =f ’ +f ”is good. So we may assume f“ = 0, that is, wijM $ for all i,j. Now let fi = h i j X iwij; we focus on ,L, which is clearly the sum of all generalized monomials o f f having label X , , . . . X , ( r -l)Xr for all n in Sym(r - 1). By symmetry, we may assume f, # 0, and we shall build “one step of a staircase,” using f,. Choose y, in M such that w t l y tq! By Corollary 1.5.3 there exist a1 = 1,a2,...,au, in F such that for each y’ in M there is a corresponding r’ in R with r’ = 0 and r’wrjy, = y’aj, 1 d j 6 u,. Let g, = ajhtj(X1,.. . ,Xr-l), and put = K + ~ ; L wijy,F. For all risuchthatriI.;’=O,l < i d t - l , w e h a v e

x;;

c.

<

zy=

gt(rl,...,rr- 1 )Y’

v’

l’,

=

C hrj(rg,. .

. 3

rr- i ) ~ ‘ a j

j= 1

C h r j ( r l ,.. . ,r,- l)r’w,jy, =,f(rl, . . .,r r - ,, r’)yrE v,. 0,

=

j= 1

Since y’ was arbitrary in M , gr is (Vo, Vi, ..., V,’_,)-valued. Now let gi = g,X,wtl. Since gr is (V,, V;, . . ., <‘-,)-valued, gi is (V,, V;, . .., v L l , T/;)valued. Moreover, g, is a sum of o, monomials, and max([y’: F]JO 6 i < t - 1 ) d u+max{u,(i < r} < u + ( D - u , ) . Hence, by induction on t , g, (and thus 9;) can be written as a sum of
L-d

0,

=

C hrjxtwtj j= 1

0,

t’t

j= 1

ajhrjxtwtl

=

C htjXr(Wtj--jWtl) j=2

(since at = l), a sum of (ut- 1 ) monomials. Hence g is now a sum of u - 1 monomials and, by induction on u, g can be written as a sum of d (ti - 1)‘ monomials each having a coefficient of rank < ( u + u - l ) + ) ( u - l)(v-2) = u+$u(u1); i.e., g is good. But u‘

-(0 -

1y

= (u - (v - 1))( u f - +v‘-2(v - 1)

+ ...+ (u - 1y-

1 )

2 u‘-

Therefore f = gi+gl can be written as a sum of d u ’ - l + ( u - 1)’ < u‘ monomials and is good. QED This result serves as the source of the entire GI structure theory. The main point, by Remark 7.1.9, is that each substitution in every generalized monomial off produces an element of finite rank, which is thus in the socle; this is the key to Theorem 7.2.9 below, to which we now lead.

260

THE THEORY OF GENERALIZED IDENTITIES

[Ch. 7

Corollary 7.2.3. Suppose f is a multilinear generalized identity of a dense subring R of End MD, f is a sum of u monomials, and F is a maximal subfield OfD. Then #can be rewritten US a sum of v' monomials with coefficients spanned (ouer F ) by the coe8cients o f f , such that each monomial has a coejicient of rank < f u ( u - 1) ouer F (and also ouer D). Proof. Noting that rank(over F) 2 rank(over D ) for any maximal subfield F of D,we pass to R F and apply Theorem 7.2.2 with all = 0 (and thus for u = 0). QED

Corollary 7.2.3 generalizes Kaplansky's theorem, because if R is a dense subring of End MD, having a polynomial identity f that is a sum of u monomials, then we can rewrite f as a sum of monomials each having a coefficient in F of rank
fi

Iff is an identity of M , ( F ) that is a sum of u monomials, then n G +v(v - I ) < f r 2 , implying u > QED Proof.

J2n.

This result can be improved considerably (cf. Exercises 2, 3, 4). Corollary 7.2.5. Suppose R is a dense subring of End M , and has a multilinear GI, f ( X . .,X l ) , which is a sum of u monomials. Then, .for every generalized monomial fn o f f and for all r l , . . .,rl in R, rank fn(rl,. . . r,) < @ + 1 ( v - 1). Proof. By Theorem 7.2.2 .fn(rl,.. . ,rl) is written as a sum of < u' terms, each of rank G f u ( u - l), so by Remark 7.1.9 rank Jn(rl ,..., r l ) < u'($u(u- 1)) = $v'+'(u- 1). QED

One could write down a proof of Corollary 7.2.5 directly and obtain rankf,(r,, . . . ,rl) 6 u'. (Detailsareleft to thereader.)However,even this bound is quite crude, and the real sign significance is that the bound depends only on u and t. Definition 7.2.6.

Iffisageneralized polynomial,GM,-(R) = u { . f ; ( R ) I

$7.2.1

The Basic Theorem

26 1

every generalized monomial o f f ) . Define Zf(R) = ideal of R generated by GM,-(R),and GI(R) = UfGM,.(R)l,fis a multilinear GI of R}. fT

Remark 7.2.7.

G I ( R ) d R . Hence GI(R) = C(If(R)lall GIsfof R).

Theorem 7.2.8. I f R is primitive, then GI(R) G soc(R); if GI(R) # 0, then GI(R) = soc(R). Proof. GI(R) c soc(R) by Corollary 7.2.5. If GI(R) # 0, then by Proposition 7.1.8 soc(R) E GI(R), so GI(R) = soc(R). QED Amitsur's Theorem

Recall f is R-proper if some generalized monomial of ,f is not a GI of R. By multilinearization, GI(R) # 0 iff R satisfies an R-proper GI. This leads to the famous theorem of Amitsur [65a] that initiated modern GI theory. A dense subring R of End M, satisfies a proper GI iff Theorem 7.2.9. soc(R) # 0 and D is PI. Proof. If R satisfies a proper GI, then GI(R) # 0, so by Theorem 7.2.8 soc(R) # 0. Moreover D has a maximal subfield F . By Corollary 7.2.3, for some nonzero x E RF, 00 > [sM : F] = [xM: D][D:F ] , implying [D: F ]

< 00.

The converse is easy. Suppose soc(R) # 0. Then there is an idempotent e such that D = eRe. If D has degree n, then S , , ( e X , e , . . . ,eX2,,e) is a GI of R, one of whose generalized monomials is e X , e X , . . . e X , , , e ; since R is prime, this generalized monomial is not a GI of R, so S,,(eX,e, . . .,eX2,,e) is Rproper. QED Improper Generalized Identities

Since the above results are so decisive, we should like to take a closer look at improper generalized identities; we shall see that they are really trivial. The key step is the linear case. Lemma 7.2.10. Suppose R is an algebra over a $eld F and f ( X l , .. ., X , ) is a multilinear generalized monomial in R { X } . I f we write ,f = r i l X l f , , f o rril in R and,/; in R { X } such that v is minimal, then the { ril I 1 d i < 0) are F-independent.

x:Y=l

Proof.

Otherwise x r = l a i r i l = 0 for suitable aiin F , not all 0, so we

262

[Ch.7

THE THEORY OF GENERALIZED IDENTITIES

may assume a ,

f=

=

1. Then

c

i= 1

c

c

C rllx1jt- C

EirilXlfl

=

1 ri1X1(j;-ai,/1)9

i=2

i= 1

contrary to the minimality of v. QED Lemma 7.2.11. I f x y = l r i , X l r i 2 is a GI of a dense subring R qf End M, with r 1 2 # 0, then { r i l11 < i d v } are F-dependent. Proof. Choose yo in M such that r12y, # 0, and let y i = ri2jf0, 1 d i ,< v.ByCorollary1.5.3thereareelementsal = 1,a, ,..., a,,inFsuchthat given y in M we have corresponding r in R with ryi = yai, 1 ,< i m. Then 0= lri,rri2)yo= x y = l r i , r v i= ( x : = l a i r i l ) y since ; y is arbitrary we get

<

(xr=

Cr=lairil= 0. Q E D

Proposition 7.2.12. Suppose R is dense in End M,, F is a maximal subfeld of D, and Z ( R ) is a f e l d such that a n y f n i t e set of Z(R)-independent elements of R is F-independent (viewed in RF). Then every multilinear generalized monomial # 0 (in R{X}) is not a GI of R . Proof. Suppose, to the contrary, that f # 0 is a generalized monomial that is an identity o f R ; we may assumef has label X 1 . . . X , and write f = x : f = l r i l X l f for l r i l in R such that c is minimal. Hence by Lemma 7.2.10 the ril are Z(R)-independent, and so by hypothesis are F-independent. Now the,f;.are nonzero generalized monomials and so by induction on t they are , r t ) # 0, and let not GIs of R. Pick r 2 , .. ., r, in R such that f 1 ( r 2 ..., ri2 =fi(r2, .. . ,r,). Then rilX,ri2is a GI of R, thus of RF, implying by Lemma 7.2.1 1 that the ril are F-dependent, a contradiction. QED

x;=

Corollary 7.2.13. I f R is simple, then every multilinear generalized monomial # 0 (in R { X } ) is not a GI of R . Proof.

of

Apply Corollary 1.5.19 to Proposition 7.2.12. QED

Corollary 7.2.14. D is a GI o f R .

lf[D:Z(D)]

=

00

and R is a D-ring, then every GI

Proof. First note by Theorem 7.2.9 that D satisfies no D-proper GI. W e claim that i f f is a GI of D, then f = 0 in D{X); the assertion follows immediately. f is a consequence of any generalized monomial of its multilinearization f . and f is D-improper, as noted above; thus we may assume .f is already a multilinear generalized monomial. Hence f = 0 in D{X} by Corollary 7.2.13. QED

In Exercise 10 we replace D by any simple ring of infinite dimension over its center, but Corollary 7.2.14 is enough for the important application to

47.2.1

The Basic Theorem

263

the fundamental theorem of generalized rational identities (cf. Corollary 8.2.12). Strong GIs Having examined R-proper GIs, we continue with R-strong G I s ;f is Rstrong iff f is proper for every homomorphic image of R. Note that f is Rstrong iff 1 E Zf(R) [since if 1 Z,(R), thenfis R/I,(R)-improper]. Hencefis (nR,)-strong for any direct product of copies R, of R.

e

Theorem 7.2.15. R is a PI-ring.

If R has a completely homogeneous R-strong GIf; then

Proof. By Amitsur’s method, it suffices to consider the case Nil(R) = 0. In this case R[I] is semiprimitive by Amitsur’s theorem. Viewing R[A] as a subdirect product of dense subrings R, of End(M,),”, YE^, for suitable vector spaces M y over division rings D,, and letting F , be a maximal subfield of D,, we view each R, E End(M,JF,; letting r, be the image of r in R,, we define B = { r E R [ I ] I{[ r y M y: F,] Iy E I-} is bounded). Clearly BaR[I]. Obviously we may throw out all generalized monomials offthat are GIs of R, so we assume all generalized monomials of f a r e not G I s of R. Also, 1 E Z,(R) E Zf(RII]),so f is R[I]/B-proper if B # R[A], in which case some multilinearization g off is R[I]/B-proper. On the other hand, g is a GI of R[A] and is thus R[L]/B-improper by Corollary 7.2.3. We must conclude that B = R, so 1 E B, i.e., {rank 1 Iy E r}is bounded, say by k. Hence S,, is an identity of each R,. Therefore S 2 k is an identity of R. QED

In the above proof, k is not bounded. Indeed, M , ( F ) satisfies the M,(F)strong GI [ e l l X l e l l , e l l X 2 e l l ]for all (arbitrarily large) n. This theorem is very useful for applications; we give one now. Theorem 7.2.16. Suupose there exists an element r in R and a polynomial g ( I ) in Z(R)[I], I being a commuting indeterminate over R , such that C R ( r )is a PI-algebra, g(r) = 0, and g’(r) is invertible (where g’ is the formal dertvative of 9). Then R is PI-algebra.

Let g(A) = x F = , o c t k A k for suitable ctk in Z(R); then g’(A) ~ = l c t k ~ ~ ~ ~ r i Note X l f for - l all - i x. in E R ( r ) ,and f(1) = g‘(r) is invertible in R. Take a multilinear R that ~ ( x ) C polynomial identity h ( X , , . . . , X , ) of CR(r).Then h ( f ( X , ) , .. . , f ( X , ) )is a GI of R, one of whose generalized monomials is f ( X l ) . . f ( X , ) ; specializing each X i -to 1, we see that 1 EI,(R), so by Theorem 7.2.15 R is a PIring. QED Proof.

= xr=lkakk12k-1. Definef(Xl) = ~

264

THE THEORY OF GENERALIZED IDENTITIES

[Ch. 7

The Modified Density Theorem and Its Consequences

Sometimes we need to use the following mild generalization of the density theorem : Modified density theorem. I f M is an irreducible R-module with D and if B is a lest ideal of R with BM # 0, then for any Dindependent elements y,, . . .,y, in M and for all y ; , . . . , y ; in M there exists suitable b in B such that by, = yi, 1 < i < m. = End,M,

Proof. BM is a nonzero submodule of M , so EM = M, and in the proof of the density theorem we could replace routinely R by B. QED

Thus, if R is dense in EndM, and R, is a subring of R containing a nonzero ideal of R , then R, is also dense in End MD. Our first use of the modified density theorem is to provide, by means of the regular representation, a general form of counterexample. Example 7.2.17 (Amitsur). Suppose R is an algebra over a field F . Viewing R as a vector space over F , we can use the regular representation of R to inject R into End,R. Let A = soc(End,R), and let R , be the F-subalgebra of EndfR generated by R and A . Then R, is dense in End,R, and soc(R,) # 0. Moreover, many properties can be passed from R to R,. For example, one can easily prove A is F-algebraic; if R is algebraic, it follows easily that R, is algebraic. A striking illustration of this method is given when we recall that Golod [64]-Shafarevich gave an example of an algebraic algebra that is not locally finite (cf. Herstein [68B]). Thus, we see instantly that a primitive algebraic algebra with proper GI need not be locally finite. Here is another application of modified density. Proposition 7.2.18. Suppose B is a left ideal of a dense subring of End M,, and BM # 0. l f f ( X I , ..., X , ) is a generalized polynomial wrth f ( B ) = 0, then f i s a GI ofEnd M,.

By modified density, given any x,,. . . ,x, in End MD and any y in ( X I , . . . , x , ) y = f ( b l , . . . ,b,)g = 0. (The point is that we have only a finite number of multiplications and additions of a finite number of elements, so we could view things in a finite dimensional subspace of M . ) Hence f ( x , , .. . ,x,)M = 0 for all x, in End MD, implyingfis a GI of End MD. QED Proof.

A4 we can find b , . . .., b, in B such that f

Corollary 7.2.19. Any primitive ring R is contained in an R-ring multequivalent to R, o f i h e form End M , for some vector space M over a suitable jield F.

47.3.1

Primitive Rings with Involution

265

Proof. Applying Proposition 7.2.18 to some closure of R, we see R is mult-equivalent to some End M,. Therefore End M, is an R-ring. QED

57.3. Primitive Rings with Involution

So far our specific study of identities of rings with involution (cf. Chapters 2, 3) has been for PI-rings that have an involution. I n order to obtain the best results, one has t o restructure ring theory so that the major concepts are redefined in terms of the involution. The obvious place to start is with “primitive,” in view of Kaplansky’s theorem. Accordingly, in this section we look at “(*)-primitive.” The key point is that we want to find some (*)analogue to injecting a primitive ring into EndM,. Now we know (by Remark 1.5.24) that a ring R is primitive iff it has a maximal left ideal A with Ann, RIA = 0. In this case we could take M = RIA, D = End, M , a division ring, and inject R into EndM, canonically. So, to find the (*)parallel, suppose (R,*) is a ring with involution and A is a maximal left ideal of R. Then A* is a maximal right ideal of R. Let M = RIA and M* = RIA* (as right module). The map (r+A) + r* + A* induces a 1 :1 correspondence from the irreducible module M to the irreducible right module M*, which we also call (*); in fact (*) is an isomorphism of the additive group structures of M and M*. Let D = End,M and D* = End M i . Remark 7.3.1. In the above notation, there is an anti-isomorphism (*) from D to D*, given by d*y* = (yd)* for all y* in M*; and there is an antiautomorphism (*) from End M, to End,.M*, given by y*P* = (by)* for all y in M (with given in End M,,). Hence (*) induces an exchange

involution of End M, 0 End,,M*, denoted as 0.

Remark 7.3.2. In the above notation, there is a natural homomorphism $: (R, *) + (End M, 0 End,,, M*, o), taking r to (r, r)(viewing r in End M, and in End,, M*);

k e r t , b = { r E R I r M = O = M*r) = ( r ~ R l r M = o = r * M ) . We are interested in the situation where ker t,b

= 0.

Definition 7.3.3. For a module M, define Ann(R,*)M= (Ann,M)n (Ann, M)*, the “largest” ideal of ( R , * ) in Ann,M. (R,*) is primitiue if Ann(,, *, RIA = 0 for some maximal left ideal A of R. Remark 7.3.4.

Ann(R,.)RIA

Proposition 7.3.5.

=

(Ann,R/A) n (AnnXR/A*).

The following statements are equivalent for (R, *):

266

T H E THEORY OF GENERALIZED IDENTITIES

[Ch. 7

(i) ( R , * )is primitive; (ii) R has an irreducible module M with Ann(,.*,M (iii) R has a primitit e ideal P with P n P* = 0.

= 0;

Proof. (i) =. (ii) is immediate from Definition 7.3.3 (with M = R / , 4 ) . (ii) s (iii) is obvious with P = Ann,M. To see (iii) (i), let A / P be a maximal left ideal of RIP such that 0 = Ann,,,,((R/P)/(A/P)) : Ann,,,(R/A). Then Ann,(R/A) G P , so Ann(R,+, R I A G P P, P* = 0. QED =j

Corollary 7.3.6. = 0.

If ( R ,*) is primitive, then ( R ,* ) is prime and Jac(R)

Usually it is more convenient to focus on statement (iii) of Proposition 7.3.5 for proofs about (*)-primitivity, but the original definition is useful when we need to work more closely with general structural properties involving (*). A (*)-Analogue of the Density Theorem

We turn now to “(*)-density.” Even for matrices, matters are quite complicated, as we saw in $2.5 while attempting to build the longest possible “staircase.” Fortunately, we could manage the computations because every involution was either of transpose type or symplectic type. In the case of primitive rings, in general, I do not know of any such classification of the involutions, but fortunately this can be done without too much difficulty when the socle is nonzero. Assume throughout that R is semiprime with involution (*). Remark 7.3.7. Suppose L is a minimal left ideal with idempotent e. Either L*L = 0 or e = eae*be for suitable a , b in R . (Indeed, if 0 # L*L = e*Re then 0 # (Re*Re)2= Re*ReRe*Re, implying, for some a E R , 0 # eae*Re, a right ideal of the division ring eRe; thus eae*Re = eRe.) Proposition 7.3.8. Suppose soc(R) # 0. Then either (i) R has u minimal left ideal L with symtnetric idempotent, or (ii) R has a minimal lejt ideal L with L*L = 0, or (iii)for some minitnal left ideal L, X*X = 0 for all x in L and End, L is afifirhi.

Proof. Case I. For some minimal left ideal L and some x in L, we have Lx*x # 0. Let a = Y*X, a symmetric element of L. By Proposition 7.1.2 L has an idempotent e with ea = ae = a. Now e*e is symmetric, and e*ea = e*a = (ae)* = a* = a, proving e*e # 0 and (e*e)2- e*eE Ann,a = 0, yielding (i). Case 11. For ever) minimal left ideal L‘ and for each .Y in L‘, LX*Y = 0. We claim that some minimal left ideal L has the following property: Property P. X*.Y == 0 for all .Y in L.

47.3.1

267

Primitive Rings with Involution

Note. If L lacks property P, then LL*

= 0.

Indeed, taking x in L such that

.x*x # 0, we have Rx*x = L, and L(Rs*.u)* = L x * x R = 0.

Now take a minimal left ideal L‘ of R , and x # 0 in L’. Then x R is a minimal right ideal of R by Proposition 7.1.6, so R.u* = (xR)* is a minimal left ideal. If Rx* has property P for some x in L‘, then we are done. Otherwise, for all x in L‘ Rx* lacks property P, so 0 = ( R x * )(Rx*)* = Rx*xR, so x*x = 0 for all x in L.!, proving the claim. So take L having property P and take a nonzero idempotent e in L. For any element r in R we have e*r*re = (re)*re = 0. Hence e*r*e+e*re = e*(r*+r)e = e * ( r * + I ) ( r + I ) e - e r * r e - e * e = 0, proving e*r*e = --*re. By Remark 7.3.7 we are done [having (ii)] unless e = eae*be for suitable a, b in R . Hence for all r I , r z in R we have e*ber,er,e

(e*r?e*b*e)r,e = -e*r$(e*b*er,e)

=

-

=

e*rre*rTe*be

=

-e*b*er,er,e

= e*berler2e,

yielding

0 = e*be[er,e,er,e]

= eae*be[er,e,er,e] = e[er,e,er,e] = [er,e,er,e];

thus eRe is a field. QED Definition 7.3.9. Suppose V is a right vector space over a division ring D. A map (,): V x V 3 D is a sesquilinear form if ( Z i ~ i , C , i ~ > d j ) = uJ)dj for all ui, c) in V and all d, in D. For Vl s V , define V: = { U E V l ( V l , u ) = 01, a subspace of V. (,) is nondegenerate if VL = 0. (,) is alterriating if ( u , u ) = 0 for all u in V. If D has an involution (*), then (,) is Hermitian when ( u , , u 2 ) = ( u z , u l ) * for all v l , v2 in V.

xi,j(vi,

Multilinearizing shows ( u , , u 2 ) + ( v z , u l )

=

0 if (,) is alternating.

Definition 7.3.10. If R has an irreducible module M with D End, M and if R has an involution (*), we say a form (,):M x M + D is (*)-compatibleif ( x y , , ~ , )= (y1,x*y2) for all x in R and y,,y, in M . =

Remark 7.3.11. is in End V,.

Given yo, y 1 E V, the map y + y o < y , , y ) (for all y in V )

Proposition 7.3.12. Suppose M is an irreducible R-module, (*) is an involution of R , and (,):M x M + End,M is a nondegenerate, (*)compatible, Hermitian (resp. alternating with End, M a field) sesquilinear form. I f X E Rsuch that xy = y o ( y l , y ) for all y in M , then X * J J = y , ( y o , y ) (resp. -yl(y,,y))for a l l y in M . Proof. For Y’ in M , (.Y*Y,Y’) = ( y , x y ’ ) = ( y , y o ( y l , y ’ ) ) = ( y , y 0 ) . (yl, y’). If (,) is Hermitian, we get ( x * y , y’) = ( y o , y)*(y‘, y,)*

268

THE THEORY OF GENERALIZED IDENTITIES

[Ch. 7

= ((Y’,YI)(YO,Y))* = (Yl(Y0,Y)~Y’) for all Y‘, implying .y*4’= Yl(Y0,J) (since (,) is nondegenerate). Theother assertion is proved analogously. QED

Theorem 7.3.13. Suppose R is semiprime with (*)and soc(R) # 0. Then there is a minimul Ieji ideal L satisfying one of‘ the following three properties (viewed as right End, L-vector space):

(1) L*L= 0; (2) L has a nondegenerate, (*)-compatible Hermitian,form ; (3) L has a nondegenerate, (*)-compatible alternating form, and End,L is a-field. In fact, we ma.v assume in Cases (2) and (3) that for any yo,yl in L therr is some r E R such thar ry = yo(vl,y) for all y in L. Proof. We apply the conclusions of Proposition 7.3.8. If (ii) holds then we have conclusion (1) here. If (i) holds then take L = Re such that e* = e (e idempotent) and define (,) by (xle, x,e) = ex:x,e, a nondegenerate Hermitian form that is obviously (*)-compatible, yielding (2); if y o = xoe and y, = xle, then yo(yl,y) = (.u,ex:)y, so we are done. Finally, suppose (iii) holds ; i.e., we may assume L = Re, x*x = 0 for all .x in L, and e = eae*be for suitable a, b in R (or else L*L = 0 by Remark 7.3.7). Define (xle, x,e) = eae*x:x2e. Clearly (,) is sesquilinear, (*)-compatible, and alternating. Moreover, if re€ Re’, then 0 = eae*Rre, so 0 = eae*beRre = eRre, implying (Rre)’ = 0, so Rre = 0 and re = 0. Thus (,) is nondegenerate, yielding (3); if yo = xoe and y1 = xle, then yo(yl,y} = (xoeae*x:)y, so we are done again. Q E D

Note that under conclusion (1) above, R cannot be prime. Example 7.3.14. In order to understand Theorem 7.3.13 better, consider (R, * ) = ( M , ( F ) , *). If (*) is the transpose, then we take e = e l l . a symmetric,rank 1 idempotent,setL = Re,,,anddefine(y,, y,) = t r ( ~ : y , ) ~F for y,, y, in L. Obviously (,) is Hermitian, nondegenerate, and (*)-compatible. The element eij of R corresponds to the transformation yweil(ejl,y); likewise eji corresponds to y + + e j l ( e i l , y ) , illustrating Proposition 7.3.12 (since eji = e t ) .

If (*) is the canonical symplectic involution, then we have n = 2m for some m and again take e = ell, but note this time, for all x in Rell, that x*x = 0. [Indeed, if x = laieil,then

c aiem+l,m+im

x* =

i= 1

i: aiem.1.i-m

i=m+l

c ai+mem+l.iy m

=

i=m+l

ai-mem+l,i -

i= 1

$7.3.1

269

Primitive Rings with Involution

so n

m

x*x =

C (-aiai+m)em+ + 1 i= 1 1,1

i=m-Cl

ai-maiem+l,l

= 0.1

Thus, we can define ( y 1 , y 2 )= t r ( e l , m + l y ~ y 2for ) , y,,y, in Re,,, and (,) is alternating, nondegenerate, and (*)-compatible. When trying to construct elements in primitive rings with involution, it is very useful to keep Example 7.3.14 in mind. Definition 7.3.15. If V is a given right D-vector space and Vl is a Dsubspace, then the codimension of V, (in V ) , written codim(V,), is [( V W l ):Dl.

If V, is a D-subspace of I/, if t = [Vl :D] < and if D is a sesquilinear form, then codim(V+)< t . [Indeed, if V, luiD, then V: 2 , ( v i D ) ’ ; for each i, clearly codim((v,D)’) = 1.1

Remark 7.3.16.

(,):V x =

I/+

n:=

Theorem 7.3.17. Suppose ( R ,*) is primitive and soc(R) # 0. There is a minimal left ideal L, and D = End, L, with a sesquilinear, (*)-compatible form (,):L x L + D having the following property for any left ideal L, with L, L # 0, and f o r anyfinite dimensional D-subspace V of L . Given y‘ in L - V and y” in ( V + y‘D)’, one has x in L , with x V = x* V = x*y’ = 0 and with xy’ = y”. In fact (fL*L = 0 for some minimal lefi ideal L, then we may take (,>to be trivial; i.e., y“ can be arbitrary in L. Proof. Using the modified density theorem we have x1 in L, such that x,y‘ = y’ and x1 V = 0. We need to find x2 such that x2y’ = y“ and x:(V+y’D) = 0; we are then done by setting x = x2xl. We shall find x2 by using Theorem 7.3.13. First assume (1) holds, i.e., L*L = 0 for some minimal left ideal L. L2 # 0, so by the modified density theorem we have some x2 in Lsuch that x,y’ = y”; x f L E L*L = 0, and we are done. Now assume (2) or (3) holds in Theorem 7.3.13, including the assertion at the end. Let L be a minimal left ideal with the desired nondegenerate, Hermitian, or alternating (*)-compatible (,):L x L + D. Then there exists some element y , in L with ( y , , y‘) # 0. Multiplying y1 by a suitable element of D, we may further assume ( y l , y ’ ) = 1. Now we have x2 in R such that x2y = y ” ( y , , y ) for all y in L. Clearly x2y’ = y”. By Proposition 7.3.12, for all y in V, xTy = p y l ( y ” , y ) for p = f 1. I f Y E V+y’D, then ( y ” , y ) = 0, so xf(V +y’D) = 0. QED

Corollary 7.3.18. W i t h the notation as in Theorem 7.3.17, given y , , ...,yc arbitrary in L, and a finite-dimensional D-subspace V of L not containing y , , there exist d , = 1,. . . ,d,, in D having the ,following property:

270

THE T H E O R Y OF G E N E R A L I Z E D IDENTITIES

[Ch. 7

For any element y in ( V + y i D)’ there exists un element r in R such that rV = r* V = 0, ryi = yd,, and r*yi = 0, 1 < i 6 1).

Proof.

Mimic proof of Corollary 1.5.3. QED

Matrix Algebras with Involution Using Example 7.3.14 as motivation, we shall now obtain an isomorphism theorem [with respect to (*)I that contains Theorem 3.1.61 and is free of restriction i)n the ring (i.e., that it contains $). Since the classification of involutions into orthogonal and symplectic types degenerates in characteristic 2, we start with a new description. Definition 7.3.19. Suppose R is primitive, with soc(R) # 0. An involution (*) o n R is Hermitian (resp. alternating) if some minimal left ideal L of R has a nondegenerate, (*)-compatible Hermitian (resp. alternating) form. (By Theorem 7.3.13 (*) must be Hermitian or alternating.) Theorem 7.3.20. Suppose R = M,(F), with two inuolufions (*), ( J ) of thefirst kind, which are both Hermitian (resp. both alternuting). Then there is a jnite-dimrnsionul extension j e l d K of’F such that, extending (*) uiid (J) naturallj’ to M , ( K ) = M , ( F ) O I . K ,we haue ( M , ( K ) , * )z (M,(K),J).

Proof. Assume (*) and (J) are both Hermitian. (A similar argument works when they are both alternating.) Take minimal left ideals L,,L, of M , ( F ) such that for u = 1,2 L, has a nondegenerate, (*)compatible (resp. (J)-compatible), Hermitian bilinear form, written (,),. By a Gram-Schmidt process. working in a suitable finite extension field K of F , we have an orthonormal base y 1”,..., ynu of L, [with respect to (,>,I, u = 1,2, i.e., ( y i u ,J , ~ , ) = d i j . Write riy’ for the element of R = M , ( K ) such that r$’y = y i , ( y j , , y ) for all y in L, (cf. Theorem 7.3.13). If ccijr$’ = 0 for r i j in K , then for each y in L, and for each i

xy,j=

n

0 = (~i,.

C

i.j= 1

n

mijrir’y)

=

C

j= 1

n

aij(.vju,Y) =

< 2 aijYju,Y), j= 1

implying each ccij = 0. Thus [r!r)ll < i,j < n} are K-independent and by a dimension count must be a base of R over K . Then it is clear that the map C a..r!?) IJ V -+ x a i j r $ ’ is a homomorphism ( R , *) + ( R ,J), which is an isomorphism because R is simple. QED

$7.4.1

Identities of Rings with Involution

271

57.4. Identities and Generalized Identities of Rings with Involution

In this section we extend the results of $7.2 to the involutory case. The theory is motivated by a much more special question asked by Herstein in response to Kaplansky’s theorem: If R is a simple ring, having involution (*), and if some classical polynomial vanishes under all substitutions of symmetric elements, is R central simple? After several years, Herstein [67] answered his own question affirmatively, and Martindale [69a] extended the result to primitive rings. Amitsur [69] found a beautiful generalization in the context of (*)-identities, that every ring having a (*)-identity with coefficient 1 is a PI-ring. In this section, we further generalize Amitsur’s theorem to (*)-GIs, obtaining Amitsur’s theorem as a special case. In order to understand better what is going on, let us lower our sights a bit, focusing on the most important part of Amitsur’s theorem, a (*)-version of Kaplansky’s theorem. The proper conjecture in the (*)-structure would be, “If ( R ,*) is primitive and satisfies an ( R ,*)-proper identity, then (R, *) is simple, and [R :Z(R, *)] is finite.” The proof would be to use some sort of density argument to build a staircase. Now we have a suitable (*)-density theorem (Theorem 7.3.17). Unfortunately, the hypotheses include “soc(R) # 0.” This condition is more related to GI-theory, so the easiest way of extending Kaplansky’s theorem to rings with involution is probably by GItechniques. This is the main step in our program, and is trivial in one special case. Definition 7.4.1. If f ( X , , X y , . . . , X , , X : ) is a generalized polynomial, then GM,(R, *) = { ,f,(R, *)[every generalized monomial .f, o f f ) , Z,(R, *) = ideal of R generated by GM,(R, *),and GT(R, *) = [GM,(R, *)I every GI .f‘ of ( R ,*)}.

u

Remark 7.4.2.

If (R, *) is special, then GI(R, *)

u

=

GI(R).

It is useful to show that a given ring with involution satisfies a special GI because then the results of $7.2 apply. Proposition 7.4.3. ( R ,*) is special.

l f ( R , * ) is primitive and R is not primitive, theri

Proof. Let P be a nonzero primitive ideal of R such that P n P* = 0 (cf. Proposition 7.3.5). Po P* is a special subring (without 1 ) of (R, *); hence, for any f ( X , , X:, . . .,X , , X : ) that is a GI of (R, * ) , f ’ ( X , , X , , . . .,X,, - X 2 , ) is a GI of P O P * , and thus of P* I( P o P*)/P, an ideal of the primitive ring RIP. By Proposition 7.2.18 , f ( X l , . . . , X 2 t is) a GI of RIP; by an

272

.THE THEORY OF GENERALIZED IDENTITIES

[Ch. 7

analogous argument f ( X , ,..., X 2 , ) IS a GI of RIP* and thus is a GI of R. QED This result, due to Baxter-Martindale [68], is one of the rare examples of an important structural result whose proof relies intrinsically on passing to rings without 1, and is very useful because it enables one to look at primitive rings, which are better known than (*)-primitive rings. Nevertheless, it is more natural not to mix categories so blatantly; we postpone the main application of Proposition 7.4.3 a little. Here is a cute corollary. Corollary 7.4.4. Zf(R, *) is primitive and R is not primitive, then ( R , *) can be injected into some mult-equivalent (End MF @ (End M,)OP,0 ) fiw a suitable vector space M over afield F. Proof. Take a nonzero primitive ideal P such that P n P* = 0. Identifying RIP* with and letting R‘ = ( R / P ) F be a closure of RIP, we can inject

( R , *) + ( R / P 0

0(R’)OP,0 ) + (End M , 0 (End M,)OP,0 ) . 0 ) -, (R’

Sinceeachidentityof(R, *)isspecial,weuseProposition 7.2.18 toseethat multequivalence holds at each step. QED Corollary 7.4.4 provides the ideal sort of injection for GI-analysis. We should now like to find a closure of primitive (R,*), in the context of the involution. This can actually be done quite easily. As in 47.3, let A be a maximal left ideal of R such that Ann(,,,,(R/A) = 0; let M = R / A and M* = RIA*, and consider the injection t,b: ( R , *) -, (End M,, 0End,, M*, 0 ) of Remark 7.3.1. Take a maximal subfield F of D. Let T = End M , and T * = EndF,M*. We define an anti-isomorphism (*) from T to T * in the obvious way ; giveii fi in T , define fi* by y*fi* = (By)* for all y* in M * . Then T O T * has an exchange involution ( 0 ) induced by (*), and (End M, 0EndDIM*,0 ) is naturally injected in T 0T * . We inject F into T O T * by taking a (in F ) to the map &, given by dCy,,y:) = (yla,a*y:). Now F is a symmetric subfield of T O T * that centralizes $ ( R ) , so we can form (t,b(R)F,o)5 ( T @ T * , o ) ;to be suggestive, we write (RF. *) in place of ($(R)F,o). M acts on the first component of each element of ( R F , * ) [viewed in ( T @ T * , o)], thereby becoming an irreducible RF-module with Ann(RF,t,M = 0. Also, as in Proposition 1.5.12, we have F = Z ( R F , *). Call ( R F ,*) a closure of ( R , *). Proposition 7.4.5. Assume ( R , * ) is primitive and not special. Then, passing to (RF.*), we have RF is closed primitive, and ( R F , * ) is multequivalent to ( R , *)

47.4.1

Identities of Rings with Involution

273

Proof. Applying Proposition 7.4.3, we see that R is primitive. Hence R is prime, implying Ann,(R/A) = 0. Now Proposition 1.5.12 applies, so R F is clearly closed. Obviously ( R F ,*) is mult-equivalent to ( R ,*). Q E D

This simple observation clarifies the GI-structure of primitive rings with involution; to begin with, we can toss away the “stumbling block,” described earlier, to extending Kaplansky’s theorem. Theorem 7.4.6. If(R, *) is prirnititle with proper GI, then soc(R) # 0. Proof. Take some multilinear,f(X,, X:, . . . , X , , X : ) which is a proper GI of R, and take primitive P Q R with P P, P * = 0. If,fis ( R , *)-special then soc(R) # 0. (If P = 0 this is Theorem 7.2.9; otherwise RIP has some minimal left ideal LIP, and then L n P* is a minimal left ideal of R.] Thus we may assumefis not ( R ,*)-special. Thus by Proposition 7.4.5 we may assume R is a dense subring of End M,, where K = Z ( R ) is a field. Moreover, we may clearly choosef’such that no subsum of generalized (*)monomials of,f’is a GI of (I?,*). Since,fis not (R,*)-special, we may then assume that ,f(X,,XT, ..., X , - , , X;C- X,,O) is not a G I of ( R , *). Thus there are elements r l r:, . . . ,r,- 1 , r:- of R such that , f ( r l ,r:, . . . ,r,- rF- I , X , , 0) is , = , f ( r , ,r:, . . . , rF- X , , X:), we see that ,fl not a GI of R. Writing . f l ( X r XF) is a proper GI of ( R , *). I ril X r r i 2+ ril X:ri2 for suitable Now write out ,f,( X t ,X:) = rij,rij in R. By Lemma 7.2.10 we may assume that { r i l l 1 < i d u ) are K-independent. Choose yo in M arbitrarily and let I/ = ri2yoK. By Corollary 1.5.3 there exist elements a, = 1, a 2 , ..., a, in K such that given y in M there is a corresponding element r in R with rrizyo = ya,, 1 < i < u. Thus, for all .Y in R such that s V = 0, we have (substituting x*r for X , )

x,Y=

x;=,

xf‘=

o=( /

c rilx*rri2+ 1 r i l r * x r i 2 ) y o u

\

i= 1

i= 1

U

=

I’

C ril.u*(rri2yo)+ C rilr*(?tr~2yo) = i= I

i= 1

1

riIx*a, (i=”l

J’.

This is true for each y in M . Hence, for every x in R such that x V = 0, we have 0 = ril.Y*ai= (xx.Y= airi:)*, implying xx,Y= airi: = 0. Thus, by the density theorem, a j r 5 ) M c V, implying air; E soc(R) and is I air:)* ;= = Cf=a i r i l ,contrary to the Fnonzero because otherwise 0 = ( independence of the r i l . Q E D

xy=

(x:Y=

We could extend Amitsur’s theorem (7.2.9) and Kaplansky’s theorem immediately at this point, but without much extra work we can even extend Theorem 7.2.2. Actually, we shall repeat the proof of Theorem 7.2.2 using

274

[Ch. 7

THE THEORY OF GENERALIZED IDENTITIES

the partial density provided in Theorem 7.3.17 in place of the usual density theorem. Suppose ( R , *) is primitive, and in fact suppose that soc(R) # 0 and R is a dense subring of End M,. Letting "subspace" denote finite-dimensional Fsubspace of M . we say f ( X , , X : , . . . ,X , , X : ) is (&)-valued for subspaces V,,. . ., if for all ri in R such that r i & = rTF = 0, 1 < i 6 t, we have ,f(r l , ..., r,)M c V,. In order to apply Corollary 7.3.18, we further assume that M is a minimal left ideal L satisfying the conclusion of Theorem 7.3.17 and (thus Corollary 7.3.18). Thus we have the nondegenerate sesquilinear, (*)-compatible form (,>:M 0 M + F.

v

.

Lemma 7.4.7 (Notation as above) SupposeJ'(X,,X : , . . . X , , X:) is a sum of' v monomials and is (v)-valuedf o r suitable subspaces V,, . . ., J( : let u = max{[ : F] 10 < i < r } . Ler W, be the F-subspace of' R F spanned by the coeficients off: Then f can be wrirten as a sum of < I + monomials with coeBcients in W,, each monomial having a coeficient of' rank < 4'"'max(u,v).

v

Proof. (Modeled after Theorem 7.2.2, with some duplications omitted.) Call a generalized polynomial good if it can be written as a sum of monomials (with coefficients in Wo), each having a coefficient of rank <4'+"max(u,v). We go by simultaneous induction on t and u, and may assume r 3 1 and 11 > 1. Write

f=

c (c h , j ( X , , X ?,..., X i * _ , , X i + ] , X i * , 1 ) . . . + c hjj(X1,x ; , .. ., xi- x:-l, xi, X?*,,, ...)X*W!. r

i t

j: 1

;=I

)XiWlj

xi - 1 .

0:

1,

V)?

1,

j= 1

where each hij is a monomial in (RfXf,.). We may assume w i j M $ & and wijM $ for all i.j by an inductive argument on v. Write& = 2;;h i j X i w i j and = I$ hIiX,?wii By symmetry, we may assume ,f, # 0. Choose y, in M such that w r 1 j ~ ,5. $ Let U = W , ~ ~ , F +w j~j y r$F~; note that [ U : F ] < u + v. By Corollary 7.3.18 there exist m 1 = 1, mz, . . . ,ac, in F such that for each y in U' there is corresponding r in R satisfying r*U = 0 = r v and rwrjyr= y u j , 1 < j < v,. Let g, = a j h I j ( X l ,X : , . . . , X , - X:- ,),and put &' = I ( + ~ > L wijyIF wijy,,F, 1 -<, i ,< t . For all ri such that ri = rT y.' = 0, 1 < i < t - I, and with y, r as in the last paragraph, we have

,

s.'

+x;i

v+x;~~

xy=

,,

v'

,

$7.4.1

Identities of Rings with Involution

275

thus there is Vd 2 V, with [ Vd : F] < 2 u + v < 3 max(u, v), such that gt(rl,rT,..., r f - l , r T - l ) M g Vd for all ri with rjv’=rF&’=O, 1 < i < t - I . Hence gt is (Vd, V;, . . ., y- ,)-valued so, by induction on t , g f is good. Let g; = g t X t w r 1 , 9 = f - g i = x:ii htjXr(wtj-Ujwt1). Clearly g; is good ; moreover, g is (Vd, V;, . . ., K’- I/;)-valued and is a sum of (v - 1) monomials and so, by induction on v , is good. Hence f = g +g’, isgood. QED

(fi+fi’)+ft’+x:yf=2

Our bound on the rank of coefficients is not as good as in Theorem 7.2.2, although with a bit more care in the proof it could be lowered from 4’+”max(u,v ) to 2’(u+[v(v- 1)/4]). It would be interesting to determine the best bound. Theorem 7.4.8. Suppose ( R , * ) is primitive with proper GI f and is not special. Then passing to (RF, *) we can rewrite f as a sum of monomials with coeficients spanned (over F) by the coejicients off, each monomial having a coeficient of rank < k over F (where k is a function of deg(f ) and of the number of monomials o f f ). Proof. Multilinearizing, we may assumefis multilinear and pass to RF. By Theorem 7.4.6 soc(RF) # 0, so we are done by Lemma 7.4.7. QED

Corollary 7.4.9. If (R,*) is primitive with a proper identity f , then either R is central simple of degree < k (where k is as in Theorem 7.4.8) or else ( R ,*) is special. Proof. If (R,*) is not special, then 1 has rank
Theorem 7.4.10.

= soc(R).

I f (R,*) is primitive and GI(R, *) # 0, then GI(R, *)

Proof. If (R,*) is special, then this follows easily from Theorem 7.2.8 and Theorem 7.1.13. If (I?,*) is not special, then R is primitive and we have GI(R, *) c soc(R) by Theorem 7.4.8, implying GI(R, *) = soc(R). QED Lemma 7.4.11. I f (R,*) is a ring with involution and R is semiprimitive, then (R,*) is a subdirect product of (*)-primitiverings. Proof. Let { P , , l y E r } be the set of primitive ideals of R. Then 0 = n { P ; . J y ~= r }n((P..nP*)lyEr},so(R,*)isasubdirectproductofthe (R/(P;. n P*), *). QED

We are ready for the main theorem of this section; cf. Rowen [76a]. Theorem 7.4.12. Proof.

I f ( R ,*) has a homogeneous strong GI, then R is PI.

B y Amitsur’s method, it suffices to consider the case NiI(R) = 0.

276

THE THEORY OF GENERALIZED IDENTITIES

[Ch. 7

Then REI] is semiprimitive and, extending (*) to RLI], we may write (R[A], *) as a subdirect product of primitive (R,,,*), y E r. Let (R),F,, *) denote a closure of (R?,*). If R,.F , is not primitive, we use Corollary 7.4.4 to embed (Ri,Fy,*) into mult-equivalent (End(M,)F70 (End(M,)F:)OP,0 ) ; if R,F;, is primitive, view R,F, as a dense subring of some (EndM,,)Fy. Letting ry be the image of r in (R,,*), define B = {~ER[A](([~;,M,.: F ; . ] l y ~ r )is bounded}. Clearly B a R[A], so B n B * d (R[A], *). Let f be a strong GI of ( R , * ) . We may assume that all generalized monomials offare not GIs of R (by throwing out all generalized monomials off which are GI5 of R ) . Also, 1 E Zf(R, *) c Z,(R[A], *). If B n B* # R[3.], thenf is (R[A]/Br-) B*, *)-proper, implying some multilinearization g off is (R[I]/BnB*,*)-proper. On the other hand, g is a GI of (R[A],*),so Zg(R[A],*) c B (bc Corollary 7.2.3 and Theorem 7.4.8); likewise g* is also a GI of (R[A], *), so Z,,(R[A], *) E B, implying Z,(R[A], *) c B*. Thus I,(R[A], *) c B n R*. We conclude B n B* = R, so 1 E B, implying R is a PIring. QED Corollary 7.4.13

(Amitsur [69]).

ff (R, * ) is PI, then R is PI.

57.5. Ultraproducts and Their Application to GI-Theory This section is spent on developing the logical tools that reduce the study of GI-theory of prime (resp. (*)-prime) rings to results on primitive (resp. (*)-primitive) rings known from the last sections. The basic idea, due to Amitsur r67] and first utilized for GIs by Martindale [72a], is to inject R [resp. (R,-*)] into some mult-equivalent, primitive R' [resp. (R'. *)I,via an ultraproduct technique. This technique is so important that we shall derive it in its full generality. Definition 7.5.1. A $her .F of a set S is a family of subsets of S satisfying the following three properties: (i) 0$ 3 ; (ii) if A , , A , €3, then A , n A 2 E 3 :(iii) i f A , E .Pand A , c A , , then A , E 9. Example 7.5.2. If S is infinite, then [ A c S I S - A is finite\ is a filter, called the Frechet3lter. Example 7.5.3. Suppose R is prime and is contained in a semiprimitive ring R , written as a subdirect product of primitive images (R,/P,,lyeT}.For any r in R , write ry for the canonical image ( r + P J of r in R1/P;.and define r, = ( y E T ( r i# 0). Then 9 = ( A G rlr, E A for some nonzero r in R'I ib a filter of r, called the Amitsur filter. [Indeed, (iii) is obvious; (i) is clear because if r: = 0 for all r, then r E njlETP,, = 0. Finally,

$7.5.1

Ultraproducts and Their Application

277

to see (ii), given nonzero a, b in R , take r in R such that arb # 0; then

r, n rb 2 r a r b E

c 9 . 1

Example 7.5.4. Suppose ( R , *) is prime and is contained in some ( R , , * ) with Jac(R,) = 0. By Lemma 7.4.11 ( R , , * ) is a subdirect product of primitive { ( R , ,* ) l y E r } . Given r in R , let r, denote the canonical image of r in R:, and define rr= {ylr; # O } . Then .F= { A E rlr, G A for some nonzero r in R } is a filter of r. [Indeed, (iii) is obvious and (i) is clear. To see (ii), given nonzero, a , b in R , we have (by Proposition 2.2.29) some element I ef R such that arb # 0 or a*rb # 0; thus r a nrb 2 rorb or I-,* n r b 2 r O r r b . But, for each y, a, # 0 iff 0 # ( a , ) *= (a*),;consequently r, = Tat,implying r, n r b E 9.1 Example 7.5.5. Let S be an algebra, and let r = {finitely generated subalgebras of S } . For each finite subset T of S , let rT= { R E F I T c R } . Clearly rTlvT, = rTl n rT,,so it follows immediately that { A E rlsome rTE A } is a filter of I-. Definition 7.5.6. Remark 7.5.7. ultrafilter.

A maximal filter is called an ultrufilter. By Zorn’s lemma every filter of S is contained in an

Proposition 7.5.8. A filter 9 of S is an ultrafilter ifl for all A either A E or ~( S - A ) E . F .

cS

Suppose A , G S and A , $ F. Let 9‘ = ( B E SIB 2 A n A , Proof. (a) for some A in S}. Obviously 9 c .F (since A , EF), so by hypothesis 9‘ is not a filter; but 9’ satisfies properties (ii), (iii) of Definition 7.5.1, so we conclude A n A , = @ for some A E 9, implying S - A , E F . (t=) Suppose 9‘is any filter containing 3, and A E F. Then S - A $ F, so S - A $ 9 ; thus, by hypothesis, A E 3.Hence .9= 9’, proving .Fis an ultrafilter. QED We are now ready for the important construction based on ultrafilters. We use the notion of “similar” (algebraic) systems from $2.6. Definition 7.5.9. Suppose S,, y E r , are sets, and 8 is a filter on r. Define the reduced product of the S , with respect to 8,written nS,/.F,to under the be the set of equivalence classes of the Cartesian product nyerSy, equivalence given by (s,) (s;) iff {y E rls, = s;} E 9. By Definition 7.5.1 is really an equivalence. Moreover, if we have sets Siyr1 < i < t + 1, y E r, and operations F , :Slvx ... x S , , -, for each y, then writing S , = nSiv/.F we get an operation F : S1 x .. . x S , + S, + defined by F ( [ ( s , , ) ] , . . . , [ ( s t , ) ] ) = [(F,(s,,, . . .,s,,))]. ( F is well defined by Definition 7.5.1.) Writing nFJ8 for F , we now can define naturally the reduced

-

-

,

278

THE THEORY OF GENERALIZED IDENTITIES

product of similar systems .Y,

=

[Ch. 7

( S l y , . . ,SfyrF,,, . . . , F k y )to be

(n.S,,/.F, . . . ,nS17/.F; nFl;./S, ... , nF,:./.F),

denoted as

n.Y,'3.

Definition 7.5.10. an ultraproduct.

If 9 is an ultrafilter o n

r, then n . Y ' ; / S

is called

The example 1 hat will interest us is that in which we take each system to be a ring R7 coupled with a faithful, irreducible module M , . Of course we want the ultraproduct to be a ring R = n R , . / S with faithful, irreducible module M = nM,,/.F. The key is the following result: Theorem 7.5.11. (LoS [SS]). Let { Y , l y E r}be a family of similar systems, and let .F be an ultrafilter of r. A formula $ holds in (n,,rccP,)/9, $ { y e T J $ holds in Yy} €3. Proof. The theorem is obvious for atomic formulas, so we argue by induction on rank (+) in Definition 2.6.2. Let Y' = nYEr.Yy.

Case I . $ = ( -I+~). Then $ holds in .Y iff $, does nor hold in 4" iff holds in ,Y'J €9(by Proposition 7.5.8) iff { Y E rl+holds in 9,.) €3. Case 11. $ = (IL, A +z). Then I!, holds in .Y iff and t j 2 both hold in 9 iff { y ~ r lholds $ ~ in .Y'y)€ 3and { y e T ( y z holds in Y,)E riT iff {;,€I-[ $, and t j 2 both hold in :fY} E 3, iff {y E rI($]A t j Z ) holds in Y',) E 3. Case 111. $ == ( 3 , ~ ~ )...,. $ ~ xi, ( ...). Then $ holds in 9' iff for some (si) E S i , $ l ( . . . , ( s i r )., . .) holdsin .(Piff {y E I- I$](. . . ,si ,. . .) holdsin .V,for some si,)E .F iff [ y er l ( 3 ~ ~ ) $ ~.(,.x.i , . . .) holds in ,YY}E .$. QED

{ y ~ r l $holds , in .'/',}#.F(by induction hypothesis) iff I - - ( y E T I $ I

Note that the property that 9 is maximal enters in the above proof only in Case I. This theorem has several important applications. including the following method of passing from prime rings to primitive rings: Theorem 7.5.12. (i) (Amitsur [67]) I f R is a prime ring contained in a semiprimitivr ring R,, then R can be injected into a primitive ring R' that satisjies all efemeiitary sentences holding in each primitive homomorphic image of R l . (ii) I f (R,* ) is prime and is contained in a semiprimitive ( R , ,*), then ( R ,*) can be injected into a (*)-primitive ring with involution that satisfies all elementary sentences holding in each primitive homomorphic image of ( R1, *). Proof. (i) L.et { P;.I y E I-} be the set of all primitive ideals of R and, enlarging the Amitsur filter on (Example 7.5.3) into an ultrafilter #, form the ultraproduct R' = ( n 7 E r { R 1 / P ) , j ) /The 9 . map T : r -+ (r?),where each rp = r P,, E R,/P.,.,is obviously a homomorphism from R to R', and ker(z)

+

47.5.1

Ultraproducts and Their Application

279

= { r E R l { y l r ) ,= O } E F } . But if r E R and r # 0, then {ylr),# O } E F , by definition of the Amitsur filter, so we cannot have rcker(z); therefore z is an injection. By definition of ultraproduct, every elementary sentence holding for each primitive homomorphic image of R, also holds for R‘. Moreover, letting M ; be a faithful, irreducible R,/P,-module, we take the ultraproduct (n,.,{R,/P,, M , } ) / 3 .Writing M‘ for ( n , s r { M , } ) / F ,we see that M‘ is a faithful, irreducible R’-module, because “faithful, irreducible” are elementary concepts (cf. Example 2.6.5). Thus R’ is primitive. (ii) Parallel to proof of (i), using the filter of Example 7.5.4 in place of Amitsur’s filter. QED

This theorem is very nice, especially in view of the fact that although “primitivity” of a ring is not intrinsically an elementary sentence, we have “made” primitivity elementary by throwing the faithful, irreducible modules into the algebraic system. Theorem 7.5.12 enables us to replace “prime” [resp. “(*)-prime”] by “primitive” [resp. “(*)-primitive”] in many assertions, and is crucial to the remainder of this chapter. Injecting Prime [and (*)-Prime] Rings into Nicer Rings

Theorem 7.5.13. (i) Every prime ring R can be injected into a multequivalent R-ring of the form End M,. (ii) If (R, *) is prime, then either (R,*) is special or else (R, *) can be injected into a mult-equivalent (R, *)-ring with involution (R‘,*) such that R’ is closed primitive. (in the latter case, R is necessarily prime.) Proof. (i) We use a series of injections, and check that multequivalence holds at each stage. First let R, = n,,,R, for an infinite number of copies R, of R, and let R, = Rl/Nil(Rl).By Corollary 1.6.26 and Theorem 1.6.21 there is a natural embedding from R into R,, and so obviously R, is equivalent to R. Now Nil@,) = 0, so by Amitsur’s theorem R2[A] is semiprimitive and is surely mult-equivalent to R,. But viewing R E RJA], we can now use Theorem 7.5.12(i) to inject R into a primitive ring R’ that satisfies every GI of each homomorphic image of R,[A] ; hence R‘ < R,[A], implying R‘ is mult-equivalent to R. But then by Corollary 7.2.19 R‘ can be injected into End M , , which is mult-equivalent to R’ (and hence to R ) . (ii) We just add (*) to the proof of part (i), inducing involutions at each stage until (R’,*), which is primitive by Theorem 7.5.12(ii). Then we conclude the proof, using Proposition 7.4.5. (Note that R is necessarily prime if R’ is primitive by Remark 2.1.39.) QED

The point of Theorem 7.5.13 is to enable us to apply Corollary 7.2.3 (or

280

[Ch. 7

7 HE THEORY OF GENERALIZED IDENTITIES

Theorem 7.4.8 i n the involutory case) as follows: Suppose R is prime and { ( X I ..... X , ) is a multilinear GI of R which is a sum of i' monomials. Injecting R into some mult-equivalent R-ring R' that is dense in End M,,, we get by Corollxy 7.2.5, for all r , , . . . , r, of R and ever) generalired monomia1.f; of,/; that rank,/;(r!, ..., r , ) d + r l + ' ( r - 1 ) (as transformation in End M " ) . Thus /,.(R) consists only of elements of bounded rank in End MF, and GI(R) E soc(R'). Let us state this result formally. Theorem 7.5.14. //' R is prime with proper GI uiid I / R is ir1;cvreti into soiiic' niulr-equrraleiit, printitioe R-ring R', theu 0 # GI(R) c R r\ soc(R').

Let us see how this theorem "works." (Jain). I t ' R is p r i i w w i t h proper GI. rlieii R Corollary 7.5.1 5 iioiizcro riil lefi or right iL1eLiI.s. [ I n partrcufar. NiI(R) = 0.1

litrs 1 7 0

Proof. By symmetry, we need only prove R has no nonzero nil left ideals. Inject R c End h l , with0 # R n soc(End :M,).Let A = R r~soc(End M,),and suppose B is ii nil left ideal. For every u i n A , [ u B M :F ] < [OM: F ] = I for suitablerdepending ona. But uBactson aBM by multiplication.so wecan view each trb as a nil mairix in M , ( F ) . By proposition 1.3.20. (tiB)'aB.Z4 = 0. Thus (uB)'+'= 0. implying Bu = 0, yielding BA = 0, so B = 0. QED

In the above proof. we used the socle to move us to a finite-dimensional setting (actually of matrices); we shall now develop this method systematically. in order to improve Theorem 7.5. I3 substantially. The main point is to note. for rER, that Ann',,r is an ideal of Rr (viewed as ring without 1 ) and to consider Rr/Ann',,r; we let - denote the canonical homomorphic image in = Rr/Ann',,r. Typical elements of Rr will be written as Ur,hr for a, b in R. Note that ii? = 0 iff rur = 0. Proposition 7.5.16. ( i ) If'Rr is prim, tlieii i s prime. (ii) ! f M is u ,fiiitl!ful (resp. irreducible) R module, theii M/Ann',r is a ,fuitl?firl(resp. irreducible) %wiodirle. -~

Proof. ( i ) If Z R r br = 0, then rarRrbr = 0, implying = 0, so 7iF = 0 or hr = 0.

rlir =

0 or rbr

~~

(ii) First note {hat (Ann;r)M c Ann',,r, so the action o f RI on R induces a well-defined module action of M/Ann',r on %. If M is R-faithful and urM E Ann;,r, then rarM = 0, so= = 0 ; thus k is Ri-faithful. If M is irreducible and 7 G M such that T is ii nonzero submodule__ of then RrT is a nonzero submodule of M , implying RrT = M , so ,GI= RrT E 'T: hence $?is %irreducible QED

a.

$7.5.1

Ultraproducts and Their Application

28 1

Call R strongly primitive if soc(R) # 0 and R is dense in End M , for a suitable vector space M over a division PI-ring D ; Theorem 7.2.9 says that a primitive ring is strongly primitive iff it satisfies a proper GI. We use Rr to characterize “strongly primitive” in terms of central simple algebras. ’ Proposition 7.5.17. then % is simple PI. (ii) If primitive and r E soc(R).

(i) !f R is strongly primitive and 0 # rEsoc(R), % is simple PI and ifR is prime, then R is strongly

Proof. (i) Suppose R is dense in End M , , and deg(D) = d. Also, suppose rank(r) = t. By Proposition 7.5.16(ii) is primitive, with faithful, irreducible module M = M/Ann’,r. But [ M : D ] = t, so & satisfies the standard identity SZd,,and is thus simple. (ii) Let L be a minimal left ideal of %. We claim that RrL is a minimal a left ideal of R with 0 # L , c RrL, then left ideal of R. Indeed, if L , is - _ _ - _ _ 0 # L , c RrL c L , implying L , = RrL = L. Thus, for every .Y in L there exists .Y, in L , such that .Y = . f l , i.e., r ( x - x , ) = 0. Hence rL c L , , implying RrL = L,, proving the claim. But this means soc(R) # 0, so by Proposition 1.5.9 R is primitive; by Theorem 7.2.2 r E soc(R).Write R as a dense subring of End M,. Clearly % has a subring isomorphic to D,implying D is a PI-ring. Q E D

Remark 7.5.18. If A G R and ~ E Athen , no ambiguity in the notation.

% = Aa/Ann’,,a,

so there is

Suppose R = End M , The GI-theory produces a cute result concerning 6. is a W-ring, for somesubring Wof R that is mult-equivalent (over W )to R. For any r in W n soc(R) we saw 6is simple PI and by Exercise 1.1 1.3 has a unit element e. Moreover, F(Ann’r) c Ann‘r so 6is an F-algebra in the natural way, and Fe c Z ( 6 ) . Proposition 7.5.19. Given the above, 0 # Z ( E ) c Fe, and Z(%) Fe. Proof. Write e = G. Suppose Wr E Z ( W r ) .Then r[.Yr, wr] = 0,for all .Y in W, implying rX,rwr-rwrX,r is a GI of W, thus of R. By Lemma 7.2.1 I rwr and r are F-dependent, so r w r q , and r x 0 are F-dependent ; thus Wr and e are F-dependent. Thus Z ( 6 ) c Fe. Note that 6 is a prime PI-ring without 1 , so Z ( 6 ) # 0. Putting W = R we get Z ( g ) E Fe, so Z ( 6 ) = Fe. QED =

Theorem 7.5.20. (i) If R is prime and satisfies a proper GI, then some central extension of R is strongly primitive and is closed (as a primitive ring). (ii) I f ( R , *) is prime and satisjies a proper GI, then either ( R ,*) is special or (*) extends to some closed, strongly primitive central extension of R.

282

THE THEORY OF GENERALIZED IDENTITIES

[Ch. 7

Proof. ( i ) Inject R into somemult-equivalent R-ring R , = End M,,and let R’ = R F E R , . By Theorem 7.5.14 we have some Y # 0 in R n soc(R,).By Proposition 7.5.19 Z(R’r) is an F-subalgebra of Fe (for the unit element e of Hence Z(R’rl= Fe, a field. k.r is thus a prime PI-ring whose center is a field, so R’r is simple, implying R’ is strongly primitive. We are done by passing to a closure of R‘. (ii) Analogus to (i), by means of Theorem 7.5.13(ii). QED

F).

Theorem 7.5.20(i ) should be called Martindale’s theorem (cf. Martindale [69,72]), although in fact Martindale’s full result was slightly stronger and will be obtained in $7.6; Theorem 7.5.20(ii) is due to Martindale [72]-Rowen [75b]. 87.6. Martindale’s Central Closure

A pretty construction, the “central closure” of Martindale [69], provided the original form of Martindale’s theorem; it has the advantage of being quite explicit, giving rise to several new theorems (and sharper statements of previous theorems). This construction works just as well for (*)-rings (and actually for rings with any finite group of automorphisms and antiautomorphisms); to keep the exposition simple, we shall mostly restrict ourselves to the case without involution, making parenthetical comments about the (*)-case and leaving greater generality to the reader. Let R be prime, and let .f = {nonzero ideals of R). [In the (*)-case,let (R. *) be prime, and let .9= {nonzero ideals of ( R , *).I Define an equivalence on = {(.L A ) I A E f and f ’ : A R is a module homomorphism: as follows: ( f , ,A , ) (.f2, A 2 ) if.f, and,f, have the same restriction to some ,4 E A , n A , in .f. Write [ j ;A ] for the equivalence class of (f, A ) . The set of equivalence classes has a ring structure, given by [ f,, A , ] + [ j i , A , ] = [.f, +f,, A , n A,] and [fl, A1][f2, A L ] = [Ill2,( A , n A 2 ) ’ ] ; we call this ring Qo(R). Let j i denote the right multiplication map,f,(r’)= r’r.

-

-+

Remark 7.6.1. There is a ring homomorphism R Qo(R) given by. r [ j i , R ] , whose kernel is { r e Rl A r = 0 for some A in . f ) = 0; in this way we view R E Q o ( R ) . -+

-+

Definition 7.6.2. If A E .f and . f : A -+ R is a module homomorphism, we say (1;A ) is admissible iff: A R is also a right module homomorphism. [In the (*)-case, we also require f ( a * ) = f ( u ) * for all Q in A.] The extended centroid Z of R is { [:JA] E Qo(R)l(,f,A ) is admissible}. -+

Proposition 7.6.3. Z(Q,(R)).

The e.utended centroid Z OJ’R is afield contaitied in

67.6.1

283

Martindale’s Central Closure

Clearly 2 is a subring of Qo(R). Suppose ( J A ) is admissible. For in Qo(R) and all ~ 1 9 in ~ 2 A n A , , f ( f i ( a i a 2 ) ) =.f’(aifi(a2)) = f(a,)f1(a2) = .f,(f(a,)a,) = f l ( f ( ~ I a 2 ) ) 7 implying “.f,fll, ( A n A d 2 ] = 0 ; consequently 2 c Z ( Q , ( R ) ) . Moreover, if [,f, A] # 0, then f ( A ) E .Iand ( k e r f ) f ( A )=f‘((kerf)A) z . f ( k e r j ) = 0, implying kerf’= 0 ; then [ J A]-’ [ f - ’ , , f ( A ) ] E Z , proving 2 is a field. QED Proof.

[.fi,Ai]

We are now ready for the key definition. Definition 7.6.4. subring R Z of Qo(R).

The ceritral closure of R, denoted by R , or RZ, is the

Remark 7.6.5. . R is a central extension of R, and Z ( R ) = 2 [since

2 G Z ( R ) c CQo,R,(R)= 21.

Remark 7.6.6. In the (*)-case, R is a (*\-ring. [Define ( z r i z j ) *= x r T z i for ri in R, zi in Z . To see this is well defined, write C:= rizi = 0 for zi = K,Ail with (fi, A i ) admissible and let A = A , ; we may assume for all a E A that Zf=&(a)ri = 0, implying for all a,, a, in A

,

1

k

C .fi(a,)r? a2 = 1jXa1rTa2) = C.lr?.f;(a,) i=1

= a , CrT./l(a*)

= q ( ~ . / i ( u ~ ) r=i f0 ;

thus

xf=l , / i ( u , ) r E~ Ann, A = 0.1

For every .Y in R there exists A in .Y such that 0 Remark 7.6.7. # A u G R. It follows easily that R is prime. [In the (*)-case, R is (*)-prime.] Thus we have transformed R into a closely related ring whose center is a field. The term “central closure” is justified by the following result. Proposition 7.6.8.

R is its owti central closure.

Proof. It suffices to show that the extended centroid of R is Z . Suppose we have 0 # A‘ a R and f ’ : A’ + R, with (,/”, A ’ ) admissible. Then, putting A = A’ P, Rand lettingf’bethe restriction of,/”to A , we have A E -9,and (,/; A ) is admissible, i.e., [,/; A ] E 2.Conversely, given admissible (,J A ) , we can definea bimodule homomorphism,/“:R A R + R, by k

I.!(1 .Yi,a,,i2) i= 1

c k

=

.Yiij(ai).Yi2?

i= 1

and (,/”, R A R ) is admissible. These two correspondences identify Z canonically with the extended centroid of R . [The (*)-case is proved analogously.] QED

284

THE THEORY OF GENERALIZED IDENTITIES

Definition 7.6.9.

[Ch. 7

R is centrally closed if R = R.

The central closure has the following remarkable property due to Martindale [69]-Bergman [74bP], generalizing Corollary 1.5.19. Recall Theorem 1.8.18. Theorem 7.6.10. J f R is centrally closed with F = Z ( R ) ,thenfor every prime extension R , 2 R we have R , R O FH , where H = C,,(R). (Note that H is not assumed to be commutative!) Proof. Otherwise we have F-independent elements x,,. . . ,.x, in H and rixi = 0 ;choose such a situation with t nonzero r l , . . .,rl in R such that minimal. 0 # r,R,x, = Hr,x,R, so t > 1. Define a mapf:Rr,R -+ Rr2R by

xi=,

f ( x u j r l bj) = xair2bj \J

for a j ,bj in R .

i

If 2.a.r J J l bJ. = 0, then letting ri = x j a j r i b j ,1 < i ,< t, we see that 0 = C:= ri.yi = C:=,rixi, implying each rf = 0 (by minimality of t ) ; in particular r; = 0. Therefore f is well defined, and obviously ( J R r , R ) is admissible. Since r2 = f ( r , ) , we get rz = arl for some a in 2 = F. Therefore r l ( s l +ax,)+ rixi = 0, contrary to the minimality of t. QED Corollary 7.6.11. I f R , is a prime, central extension of centrally closed R , then R1 = R O Z c R , Z ( R , )If. a,,. . .,a, are Z(R)-independent elements of Z ( R , ) and x i = l r i a i= Ofor ri in R, then {r,, ...,rt) are Z(R)-dependent.

Now we look at some examples that show that the central closure is a good non-PI analogue to the ring of central quotients of a prime PI-ring. Example 7.6.12. Every simple ring is centrally closed. [Indeed, R is the only nonzero ideal of R, and every admissible V; R ) can be identified with the element ~ ( I ) E Z ( R J . ] Example 7.6.13. If R is prime PI, then the ring R , of central quotients of R is central simple, from which it follows quite easily that R , = R.(Proof is left for the reader.) In this case Theorem 7.6.10 is part of a famous theorem of Wedderburn (cf.Jacobson [64B, p. 1181).

Wedderburn's result just quoted is needed for Exercise 7. Example 7.6.14. Suppose R is a dense subring of End M,. There is a canonical injection J / : Z -+ Z ( D ) .[Proof. Take 1.1; A ) admissible, noting that A M = M and Ann', A = 0 because 0 # A a R and M is faithful, irreducible. Thendefinef': M -+ Mby,f"(ay) =f(a)yforanyain A,yin M.Ifa,y, = a2y2, then for each u in A, 0 =f(a)(a,y, -a2y2) = f ( a a l ) y ,-,f(aaz)y2 = a(f(al)yl -,f(a,)y,), so ,f'(alyl )-f'(u2yz) E Ann', A = 0, proving .f' is well

57.6.1

Martindale’s Central Closure

285

defined. Moreover, clearly f ’E End, M = D, and f ’ commutes with all elements of D, so f ’ E Z(D). We define $ :Z + Z(D) by $([A A ] ) = f’; it is immediate that $ is a well-defined injection.1 Thus, identifying Z(D) with Z(End M , ) canonically, we see that R c RZ(D) G End M,. In particular, if Z(End M,) c R, R must be centrally closed. A special case of this result is that every closed primitive ring is centrally closed, but the converse is far from true. (If D is not commutative, End M , is centrally closed but is not closed as a primitive ring.) This example gains significance from the following theorem : Theorem 7.6.15. (i) If R is prime with proper GI, then the central closure R is strongly primitive, with nonzero socle. (ii) ’1 (R, *) is prime with proper GI, then either every GI of’(R, *) is special or else, letting ( R , *) be the central closure of (R, *), we have R is strongly primitive with’nonzero socle. Proof. By Martindale’s theorem we can embed R in some closed, strongly primitive central extension R’. By Theorem 7.5.14 there exists nonzero r in R n soc(R’). Now let F = Z ( R ’ ) and consider R’r = R’r/Annk.,r, a central simple Fe-algebra by Propositions 7.5.17 and 7.5.19, where e is the multiplicative unit of R’r. Now 6 can be viewed as a 2-algebra without 1. Nevertheless, setting t = [%:Fe] 1, we claim every t elements of 6 are Z-dependent. (Indeed, if r l , . . . ,rr are elements of R, then rTr,. . ., are F-dependent, implying rrl r , . . . ,rr‘r are F-dependent, and so rrl r, . . . ,rr,r are Z-dependent, implying cr,. . . , are Z-dependent.) But % is a prime PI-ring without 1, and thus has a nonzero central element c. Since {c, . . .,c’} are Z-dependent, we have aici = 0 for suitable q2 1 and ai in 2 such that aq = e. Hence a i C q = e, so eE Rr. Now Rr is finite dimensional over Z e , so 6 is simple. Thus R is strongly primitive by Proposition 7.5.17. (ii) Proved analogously to (i). QED

+

xt=

Actually, there is a formulation ofTheorem 7.6.15(ii)that lies more correctly in the (*)-theory;cf. Exercise 8. Theorem 7.6.15(i) is the full formulation of Martindale’s theorem. The central closure is a very natural construction, and has been used by Bergman r74bPI to prove INC for finite extensions (cf. Exercise 11); there are probably many more uses for it, still to be discovered. Let us conclude this chapter by reformulating Theorem 1.4.34 in a very general setting. Actually, we shall present a magical proof that is completely GItheoretic, independent of Lemma 1.4.33! Theorem 7.6.16. For any prime ring R with central closure RZ, elements r l , . . . , I , of R are Z-dependent i$ C 2 ‘ -l ( r l r ...,rtr X , , . . ,XZtis a GI ofR.

286

THE THEORY OF GENERALIZED IDENTITIPS

[Ch. 7

Proof. Passing to R Z , which is mult-equivalent to R, we may ;issume R is centrally closed, with Z = Z ( R ) . If r l , . . .,r, are Z-dependent. then C z f - l ( r l ,.., . r,, X i , I , . . ., Xz,-l) is a GI of R since C z f - is I-normal. Conversely, suppose C2,- l ( r l . . . . ,r,, X,,l , . . . ,X Z I - is a GI or R . Itiject R into some mult-equivalent R-ring End M,; (Theorem 7.5.I.3). Then Czf-l(rl,...,r,, X , + l,:. . ,X z f - is a GI of End M F .But

CZ,-l(rl,. ..,r1. x,+1,. ... xz*1) f

=

C ( - t ) i + l r i X , + l C Z , - . l O,..., . l ~ ; - ~ , r,...., ; + r i , ~ i ,t . L. . , x ~ , - , ) .

i= 1

By Lemma 7 2.1 1 either r l . . . . , I ' , ;ire F-dcpciideiil o r each Czf-3(rl,.... r i - l , r i + r , , X r i z,..., X 2 , , ) is a ( ; I . in which case. by induction on L, r l , . . ., ri- rit . . ,rf are F-dependent, so certainly r l , . . ., rf are F-dependent in either case. But then by Theorem 7.6.10 r l , . . . ,rf are Z dependent. QEC)

,,...,

EXERClS ES

87.2 1. If R has a proper GI, then R has a proper multilinear GI ofdegrce 2. ( H i i l l : You may assume that R has a proper GI with the property that no subsum ofgenerali7c.d moiiomials isa GI of R ; then specialize.) This fact can be used to simplify several proofs. 2. Using the notation and hypotheses ofTheorem 7.2.2.prove that,/can be wriltrn iis a sum ofmonomials with coefficientsin W,one ofwhich has a nonzero coefficient ofranh ::U + rr(log, L,), where n = [W: f]. (Hirrt: You may assume ti, < $.) 3. Every multilinear identity of M,(F) is a sum of at least 2k monomials. *4. Can the bound o f Exercise 3 be improved? [A plausible conjecture based on S,, is (2k)!] 5. IfS,(a,X, ,..., rr,X,)isa GI o f R = E n d M , , t h e n e a c h a , E s o c ( R ) . 6. IfR isdense in End M,, where D has degreed, and if L is a sum of u minimal left ideals o f R , then S,,+l is a n identity of L . 7. IfA,Bareleft idt-alsofasemiprimeringR,havingrespectivepl-degrees u , ~thens,,,,,,,. , is an identity of the left ideal A + B ; in fact, GI(R) = x{left ideals of R that satisfy a standard identity). 8. In any ring, the sum of two left ideals that are P1 is PI. 9. If R is simple with R-proper GI. then R is PI. (As usual, I E R . ) 10. If R is simple and [R:Z(R)] = xj, then R is equivalent to every R-ring containing R. 11. If R is semiprime with f and satisfies the GI [r.[r.X,]]. then rEZ(R).

,

87.4 1. If (R, *) has ii symmetric (resp. antisymmetric) element r satisfying a polynomial y(L) in Z(R)[L] such that C , ( r ) is PI and y'(r) is invertible, where g' is the derivative of y. then R is PI. 2. If ( R , * ) IS semiprime with and satisfies the GI [r.[r.Xl +X:]] with r* = r. then r E Z ( R , *). (This exercie is useful in studying Jordan algebras.)

Ch. 71

Exercises

287

$7.5 I . Suppose R is prime [resp. (R, * ) is prime] and has a homogeneous R-polynomial/that is central [resp. (R, *)-central]. Then R is a PI-ring whose PI-degree is bounded by a function of the number of monomials off; or,fis constant. 2. Suppose ( R , *) is prime,fI ( X , , . . . X,) is a homogeneous, generalized (*)-polynomial,and j i ( X , + I ,.... Xu)is an ( R , *)-proper, classical (*)-polynomial such that [ is a GI of(R,*). Then either (i),/, is a constant in Z(R, *), ( i i ) / I is a GI of (R, +), or (iii) R is a PI-ring [ofdegree bounded by a function on deg(,f,) and the number ofmonomials off,]. (Hirrr; Prove the result first for the special case ,I; is a constant and obtain the general result through specializing using Exercise I.) 3. Generalize Corollary 7.5.15 to rings with involution. 4. Using ultrafilters, inject an algebra R into an algebra R' such that every elementary sentence holding for all finitely generated subalgebras of R also holds for R'. Generalize this to arbitrary algebraic systems. 5. Here is a fact useful in the papers of Passman [71b, 72a. 72b] on group rings with polynomial identity. If every finitely generated subgroup of a group G has an abelian subgroup of finite index
.

-

1. Suppose R is prime with proper GI, and 0 # X E S O C ( R )then ; Z ( R z ) :Z . Hence, for B R, there is a in Z such that ax E B. 2. If R is prime, then every completely homogeneous, R-proper GI of R is a GI of R .

a

Here is an outline of the structure theory of semiprime rings with GI ; cf. Rowen [77b]. If R is semiprime, call R a GI-ring if GI(R) is a large ideal of R.

3. Every semiprime GI-ring has no nonzero nil left or right ideals and is left and right nonsingular. 4. If R is a semiprime GI-ring and satisfies ACC(left annihilators), then R is a PI-ring and the ring of central quotients of R is semisimple. 5. A primitive GI-ring with ACC(ideals) need not be PI. 6. Say R has bounded iridrz k i f ? = 0 for every nilpotent r E R. Any algebraic, semiprime GIalgebra of bounded index is PI. ( H i n t : Pass to the prime case and then to the central closure.) 7. (Montgomery-Smith [75]) I f R is an algebra over a field F and .4 E R is finite dimensional over F (not necessarily with the I o f R ) such that A 0 ,K is semisimple, where K is the algebraic closure of F , and if C , ( A ) is a PI-ring, then R is PI ; if, moreover, R is semiprime, then PIdeg(R) is bounded by a function of PI-deg(.4) and PI-deg(C,(A)). [E\-terisitv h i r l r : In rapid succession, we may assume F = K , Nil(R) = 0, and R is prime: letting e bea central idempotent of .4 such that eAe is simple, we have S,, is an identity of e,4e for some d and eRe = eAe @ Z,eRe,C,,,(eAe). If C , ( A ) satisfies an identity of degree t , then D , , ( e X , e , .. . ,eX,,e) is a G I of R, so we conclude by using theorems from 657.2 and 7.5.1 8. I f (R,*) is prime with proper GI. then the central closure (R,*) is primitive, and GI(R, * ) G soc(R, *); moreover, if M is an irreducible R-module with AnnlR,*, M = 0. then deg(End, M ) is finite.

288

THE THEORY OF GENERALIZED IDENTITIES

[Ch. 71

9. Prove Theorem 7.6.16 using Lemma 1.4.33 instead of Lemma 7.2.1 I . 10. (Bergman [74bP]) I f S is a finitely spanned extension of a prime ring R, then R E

s

canonically. 11. (Bergman [74bP]) INC holds for all finitely spanned extensions ofarbitrary rings. (This result generalizes Theorem 4.1.8.) [E.
5s ,