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1 This article was downloaded by: [Weizmann Institute of Science] On: 14 April 2011 Access details: Access Details: [subscription number ] Publisher Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Linear and Multilinear Algebra Publication details, including instructions for authors and subscription information: Nill -polynomials in rings A. Giambruno a ; A. Valenti a a Dipartimento di Matematica, Università di Palermo, Palermo, Via Archirafi 34, Italy To cite this Article Giambruno, A. and Valenti, A.(1989) 'Nill-polynomials in rings', Linear and Multilinear Algebra, 24: 2, To link to this Article: DOI: / URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Linear and Multilinear Algebra, 1989, Vol. 24, pp Reprints available directly from the publisher Photocopying permitted by license only Gordon and Breach Science Publishers, S.A. Printed in the United States of America Nill *-polynomials in Rings A. GIAMBRUNO AND A. VALENTI Dipartimento di Maternatica, Universita di Palerrno. Via Archirafi 34, Palerrno, ltalv (Received February 25, 1988; in final form June 8, 1988) Let R be an algebra with involution * over a field F of characteristic different from 2 and let f(x,,..., x, x:,..., x,*) be a multilinear *-polynomial over F which, when evaluated in R, takes nilpotent values. We prove that if R has no nil right ideals and IF1 > 5, then f is a *-polynomial identity for R. 1. INTRODUCTION Let F be a field of characteristic different from two and R an F-algebra with involution *. In this paper we study multilinear *-polynomials that take nilpotent values in R. We shall prove that, under suitable hypotheses on R, such polynomials are *-polynomial identities for R. Let X = {x,,..., x,,...) be a countable set of unknowns and F{X, *) = Fix,, x:,..., x,, x:,...) the free algebra with involution in the xi's and xf's. The elements of FIX, *) are called *-polynomials. A *-polynomial f (x,,..., x,, xt,..., x,*)e F{X, *) is multilinear if, for each i (1 < i < n) either x, or x:, but not both, appears in each monomial off. If R is an F-algebra with involution * we say that a *-polynomial f(x,,..., x,, xt,..., x:) is nil valued in R if for all r,,..., r, R ~ there exists an integer m = m(r,,..., r,,) 2 1 such that f (r,,..., r,, rt,..., r,*)m = 0. We shall prove that if F has more than five elements and R is a ring with no nil right ideals then every multilinear *-polynomial nil valued in R is a *-polynomial identity (*-PI) for R. One could ask if this result is still valid if one replaces the hypothesis "no nil right ideals" with its two-sided version "no nil ideals". Unfortunately the following two special cases are still open: on one hand it is a long-standing question of Herstein whether, in a ring with no nil ideals, xlx2 - x2x, nil valued implies x,x, - x2x1 a polynomial identity for R (i.e., R is commutative) (see [7, p. 8001); on the other hand if x, + xt is nil valued in R (or more generally if all the symmetric elements are nilpotent) it is an open question of McCrimmon whether this forces R to have some non-zero nil ideal (see [3, p. 831). However, both these questions have a positive answer if R has no nil right ideals. We remark that if R is a ring and f is a multilinear polynomial (not a *-polynomial) an analogous theorem was proved in [I]; also, if we set [a,, a,] = a,a2 - a2a, and, The research of the first author was partly supported by a grant from MPI, Italy. 127

3 128 A. GIAMBRUNO AND A. VALENTI more generally, [a,,..., a,] = [[[a,, a,], aj,..., a,], then in [2] and [6] it was proved that if R has no nil right ideals and [x, + x:,..., x, + x,*] is nil valued in R then [x, + xt, x, + x:] is a *-polynomial identity and R satisfies the standard identity of degree four. Throughout this paper F will be a field with more than five elements, char F # 2 and R will be an F-algebra with no nil right ideals. S = { x R ~ 1 x = x*) and K = (x E R 1 x = -x*) will be the sets of symmetric and skew elements of R respectively. Also, f(x,,..., x,, x:,..., x,*) will be a multilinear *-polynomial nil valued in R. Let Z, = {I, *) be the group with two elements, S, the symmetric group on n symbols and H, = Z, - S, the wreath product of Z, and S,. It is clear that we may assume (as we will) that f is of the following form xi, if g, = 1 (g,o)=(gl,...,gn; 0) and xs1= x*, if g, = * SIMPLE ARTlNlAN RINGS In this section R will be a simple artinian ring. By Wedderburn's theorem R M,(D). the ring of k x k matrices over a division ring D. In R are defined two different types of involutions [4, p : (1) the symplectic type involution: in this case D = F is a field, k is even and * is given by (Aij)* = (A;) where the Aij's are 2 x 2 matrices over F with involution given by (2) The transpose type involution: let -: D + D be an involution in D and X = diag(c,,..., c,) G M,(D) such that c, = ci, for all i. If A = (a,,) E MJD), then * is given by A* = (alj)* = X(ii,,)X - l. Let eij (i, j = 1,..., k ) be the usual matrix units of M,(D). Given a sequence u = (A,,..., A,) of matrices in M,(D), the value of u is the product lul= AIA,...A,. Also, if (g, a) H, we write u(~-")= (A:(,,,..., A;;,,) where the symbols have the meaning mentioned above. If a,,..., a, E D, the sequence u = (a, e,, j,,..., a, eircj,,) is called simple. A simple sequence u is even if there exists (g, o ) H, ~ such that lu'g,")i = be,, # 0, for some b E D; u is odd if ~ u'~~"'~ = beij f 0, for some (g, a) H,, b~ D and i # j. Now, given a simple sequence u and an integer t, 1 < t < k, we define o(u, t) to be the total number of occurrences of t as a left or right index of the unit matrices eimjm appearing in u. It is clear that if u is simple, then (ul = be,, # 0 only if o(u, t) is even for all t, 16 t f k ; also, (u( = beij # 0, i # j, only if o(u, t) is even for all t # i, j and o(u, i), o(u, j) are odd.

4 *-POLYNOMIALS IN RINGS The basic fact about simple sequences is the following LEMMA 1 Let u E M,(D) be a simple sequence and (g, o) E H,. Then (1 ) if I U I = aeii, then lu(g,")l = bejj for some b E D, 1 6 j < k. (2) If IuI = aeij, i # j, then for some b, CE D u ( q, ~ ) l = {;'ij + ceji$ if * = transpose type e,,, t # m, if * = symplectic type. Suppose * = transpose type. Since for all t, ~(u'~~"', t) = o(u, t), by the above remark the conclusion of the lemma is clear in this case. Suppose now * = symplectic type. Recall that the involution * acts in the following way on the matrix units: e l l if i=odd,j=even +l i+t, if i, j odd \ej-i i-13 if i, jeven. Now, suppose that eij occurs once in the simple sequence u and assume for instance that i is even and j is odd. If u' is the sequence obtained from u by replacing eij with e;, then in u' four indices, namely i, i - 1, j, j + 1 have changed parity. From this fact it is easy to see that in u'~."' the number of indices that change parity is always a multiple of four. The conclusion of the lemma now follows easily. The previous lemma allows us to compute the value of a *-polynomial on a simple sequence. In fact we have LEMMA 2 Let u E Mk(D) be a simple sequence. Then k (1) if u is even, f (u, u*) = ccieii; i= 1 (2) if u is odd, for some a, b, aij D Write aeij + beji, if * = transpose type f&, u*, = { ' aijeij, if * = symplectic type. i#j and an application of the previous lemma completes the proof. For the proof of the next lemma it is crucial that f is a nil valued *-polynomial.

5 130 A. GIAMBRUNO AND A. VALENTI LEMMA 3 If f (u, u*) = 0 for all even simple sequences u, then,f is a *-polynomial identity for Mk(D). have Since f vanishes on even simple sequences, for all A,,..., A, MJD) we f(a1,...,a,, AT,..., A,*)=zaif(ui,u:) where the ui's are odd simple sequences. Therefore in order to prove the lemma it is enough to prove that f (u, u*) = 0 for all odd simple sequences. We distinguish two cases : * = transpose type. If u is odd, by the previous lemma f (u, u*) = aeij + beji, Case 1 for some ifj and a,b~d. Now, if a# 0, b# 0 take m> 1 such that 0 = f *"(u, u*) = (ab),eii + (ba),ejj. It follows ah = ba = 0 and this is a contradiction. Suppose then that f (u, u*) = aeij, a # 0. Let y ED be such that ycj!c;' # 0, i 1; such an element exists since I F / > 5. If d = cjjcl1 and I = unit matnx, the element is a unitary element (that is v* = v-') of Mk(D). Let be the inner automorphism of Mk(D) induced by v. By acting diagonally, q, induces an automorphism on M,(D) X-x Mk(D); let uq be the image of u under this tl automorphism. Then, by the hypotheses placed on f, f(uq, u*~) is a sum of odd simple sequences and, so, its diagonal entries are all zero. We have: f ( u~,!i*") = f (u, u*)"' = vj-(u, u*)v* and, equating to zero the coefficient of eii we get hence a = 0 and the result is proved in this case. Case 2 * = symplectic type. By the previous lemma, if u is an odd simple sequence, We have to prove that aij = 0, for all i, j. Now, if t # m are even, the matrix f (u, u*) = C ai.ieii. i f j is a unitary element of Mk(D). As above, computing explicitly the following expression f (uq, u*") = f (u, u*)'+' = ( 1 + e,, - ern_,,-,) C aijeij(l - elm + ern-,,-,) and equating to zero the coefficients of e,, and em_,,-, we get a,, = a,-,,-, = 0. i# j

6 i Now, similar calculations using the unitary matrices and when t is even, m is odd and t is odd, m is even respectively, yields the desired conclusion. We can now prove the final result for simple artinian rings. THEOREM 1 Let R be a simple artinian ring. If f is a multilinear *-polynomial nil valued in R then f is a *-polynomial identity for R. By the previous lemma it is enough to prove that f (u, u*) = 0 for all even simple sequences u. Let u be such a sequence. Then f (u, u*) = 1 aieii and, since f (u, u*) is nilpotent, this forces ai = 0, for all i. Thus f (u, u*) = GENERAL RINGS Recall that R is a ring with no nil right ideals and f is a multilinear *-polynomial nil valued in R. Our approach to this general situation will be that of looking at prime images of R satisfying a generalized polynomial identity (briefly GPI). In fact, in the presence of a GPI it is possible to prove the result for prime rings as the following lemmas show. Recall that by Posner's theorem [3], if R is a prime PI-ring then its ring of central quotients Q is a central simple algebra of dimension m2 over its center and one defines PI-deg(R) = m. LEMMA 4 If R is prime and satisfies a generalized polynomial identity then f is a *-polynomial identity for R. By [3, Corollary to Theorem either R satisfies a PI or for every integer m > 1, there exists a prime PI*-subring R'"' of R such that PI-deg(R(")) 3 m. If R is PI then R is an order in a finite dimensional central simple algebra Q and the involution * can be extended to Q in a natural way so that f will still be nil valued in Q (see for instance [2]). But then by Theorem 1 f is a *-polynomial identity for Q and so for R. Suppose now that for every m 3 1 there exists R'"' c R, prime PI*-subring and PI-deg(R'"') 3 m. In this case, since f is nil valued in R'"', by the first part of the proof f is a *-PI for R'"', for all m. But then by 13, Lemma R'") satisfies a PI of degree 2 deg( f ); so m d PI-deg(R'"') d def( f ), a contradiction. LEMMA 5 Let P be a prime ideal of R such that RIP satisfies a generalized polynomial Identity. Then for all r,,..., rn E R, f (rl,..., rn) E P and f (r,,..., rn)* E P. Suppose first that P* c P. Then RIP has induced involution and f is nil valued in RIP. By Lemma 4, f is a *-PI for RIP and the lemma is proved in this case. If P* Q P, (P + P*)/P is a two-sided ideal of RIP and, since RIP satisfies a GPI,

7 132 A. GIAMBRUNO AND A. VALENTI RIP, and so (P + P*)/P, has no nil right ideals (see [6, Lemma 23). Now write is the sum of all the monomials of f in which the variables xf',..., xin appear. It is clear that if g = (g,,..., g,) E B; then, for all ry',..., r,9"~ P*, Hence f, is a polynomial (not a *-polynomial) nil valued in (P+ P*)/P. Since (P + P*)/P has no nil right ideals, by [I, Theorem 31 f, is a PI for (P + P")/P. But then, since by [3, Theorem # Z(P + P*/P) c Z(R/P), it can be easily proved that f, is also a PI for RIP. It follows that f (rl,..., r,) = 2 fg(ry1,..., r,"") E P. 9 ~ 2 3 This same argument applies to the polynomial f (x,,..., x,)* leading to the desired conclusion. The following result which holds for general rings will be used later. LEMMA 6 Let A be a ring with no nil right ideals and a A such that a2 #O. Let B = uaa and N = {b E B / aba = 0). Then BIN has no nil right ideals. Let p be a nil right ideal of B = BIN. In order to prove that p = 0 we may clearly assume p = 3B. For b E B let m 3 1 be such that (xb)" E N. Then a(xb)"a = 0. If x = aza and b = ara we get a(aza2ra)"a = 0; hence (a2za2r)"+' = 0, i.e., a2za2a is a nil right ideal of A. By the hypotheses on A, a2za2 = 0; thus x = aza E Nand 5 = 0. LEMMA 7 Suppose f is not a *-polynomial identity for R. If y,,..., y, E R are such that = f (yl,..., y,)"- l # 0 and f (yl,..., y,)" = 0 then yy* is not nilpotent. Suppose that yy* is nilpotent. Since y2 = 0, y + y* and y - y* are both nilpotent. Now if y + y* # 0 take t > 1 such that s = (y + y*)'-' # 0 and (y + y*)' = 0; if y - y* # 0 we let t > 1 such that s = (y- y*)'-' # 0 and (y- y*)' = 0. In any case ~ Eor S K,s#O and s2=0. Let r,,..., r, E R and let m 2 1 such that f (sr,,..., sr,)" = 0. Since s2 = 0 we get = g (sr,,..., sr,)"s where g(xl,..., x,) is a polynomial (not a *-polynomial). So, if lr(s) is the left annihilator of s in R, the above equalitfays that g is nil valued in the ring Since R, has no nil right ideals, by [I, Theorem 31 g is a PI for R,; hence g(sx,,..., sx,)s is a GPI for R.

8 Let now P be a prime ideal of R such that s$p. Then s= s + P # 0 and, so, g(sx,,..., Sx,)S is a GPI for l? = RIP. By Lemma 5 it follows that f(y,,..., ~,)EP and f (y,,..., yn)* E P which imply y, y* E P. Hence s = (y k y*)'-' E P, and this is a contradiction. We are now in a position to prove our result in its full generality. THEOREM 2 Let R be a ring with no nil right ideals. If f is a multilinear *-polynomial nil ualued in R, then f is a *-polynomial identity for R. We will prove the theorem by induction on n = def( f ). Suppose first deg( f ) = 1 ; then f = f (x, x*) = x + ax*. If a # - 1, then for all r E R, r E R}, and f (r + r*) = (1 + a)(r + r*); hence, since f is nil valued we get that {r + r* / so S is a nil set. Suppose that f is not a *-PI for R. Take O # SES such that s2 = 0 and let r E R. Then, if m 3 1 is such that (sr + r*sjm = 0, we get 0 = (sr + r*s)"s = (sr)"~. This says that Rs is a nil right ideal of R; hence s = 0 and this is a contradiction. On the other hand, if cc = - 1, f (x) = x - x*; thus (r- r* 1 r E R), and so K is a nil set. As above this leads to a contradiction also in this case. We assume now that deg( f ) > 1. Suppose that f is not a *-PI for R; let y,,..., ~,ER be such that Y = f(y,,..., y,)"-' # 0 and f (y,,..., y,)" = 0, for a suitable m > 1. Then y2 = 0 and by Lemma 7, yy* is not nilpotent. Let r,,..., re-, E R; since y2 = 0 then for a suitable m 3 1 we have where h(x,,..., x,-,) is a non-zero *-polynomial. Set a = yy*; let B = ara and N = {be B 1 aba = 0). Clearly B has induced involution and N = N*. Also, by Lemma 6, BIN has no nil right ideals. By what we proved above we have that f ~ all r r,,..., rn-, E R, m = m(r,,..., rn-,) 3 1. This says that h(x,,..., x, -,) is a "-polynomial nil valued in BIN. Since deg(h) < deg( f ), by the inductive hypothesis we have that h is a *-polynomial identity for BIN. Thus ah(ax,a,..., ax,_,a)a i$ a *-GPI for R. Let now P be a prime ideal of R such that a2 $P. If P* c P, then RIP has the induced involution and it satisfies the same *-GPI of R. By [5, Theorem 71 then RIP satisfies a GPI; but then by Lemma 5, f (y,,..., y,) E P and, so, a2 = ( ~y*)~ E P, a contradiction. If P, then as in the proof of Lemma 5, we can obtain a GPI for (P + P*)/P and, so, for R. By Lemma 5, f (y,,..., ~,)EP and, so, a2 E P, a contradiction. References [I] B. Felzenszwalb and A. Giambruno, Periodic and nil polynomials in rings, Can. Math. Bull. 23 (1980), [PI A. Giambruno, Algebraic conditions on rings with involution, J. Algebra 50 (1978),

9 134 A. GIAMBRUNO AND A. VALENTI 131 I. N. Herstein, Rings with Inr~olutron, University of Chicago Press, Chicago, N. Jacobson, Lectures on Quadratic Jordan Algebras, Tata Institute, Bombay, [5] C. Lanski, Differential identities in prime rings with involution, Trans. Amer. Math. Soc. 291 (1985), [6] A. Valenti, A note on rmgs with involution, Boll. UMI (7) 1A (1987), [7] F. VanOystaeyen (Editor), Ring Theory, Proceedings of the 1978 Antwerp Conference, Marcel Dekker, New York, 1978.