• Abstract and Figures
• Public Full-text

PROCEEDINGS OK THE AMERICAN MATHEMATICAL SOCIETY Volume K7. Number 4. April I'M A-INNER AUTOMORPHISMS OF FILTERED ALGEBRAS. II SUSAN MONTGOMERY1 Abstract We commue earlier work and compute the A'-inner automorphisms of the ring of differential polynomials in one variable over an arbitrary domain. This is then applied to iterated Ore extensions. We also show that the ring of generic matrices has no nonidentity automorphisms which fix the center. In this paper we continue the study of A'-inner automorphisms of filtered algebras begun in [7]. which will be referred to as (I) in what follows. In (1) we described the A-inner automorphisms of the enveloping algebra of a Lie algebra, and of the ring of differential polynomials A = R[x; d] in one variable over a commutative domain R. Here we first show that the ring of generic matrices has only the trivial A-inner automorphism, equivalently, any automorphism fixing the center is the identity automorphism. We then extend our result from (I) on differential polynomials, allowing an arbitrary prime ring R as the coefficient ring. When R is a domain, we explicitly determine the group of A'-inner automorphisms of A = R[x; d] in terms of R and d. We then apply these results to certain iterated Ore extensions; in particular, if the original coefficient ring is commutative, then the group of all A'-inner automorphisms is abelian. In what follows, A will always denote a prime ring. Recall that o G Aul(A) is X-inner if it becomes inner when extended to the (right) Martindale quotient ring Q0( A) of A; when A is an Ore domain, o is A'-inner if and only if it becomes inner on the quotient division ring D of A. We shall need the following properties of Q0(A): it is the (right) quotient ring of A with respect to the filter l.'fof all nonzero two-sided ideals (that is. Q0(A) = lim -Hom^/^, A)), it is a prime ring with center C — C( A ) a field, called the extended center of A, and A may be imbedded in Q0(A) as left multiplications. By construction, for any 0 ¥= x G A, there exists a nonzero / of A so that 0 g* xl G A. Fundamental in what follows is an internal characterization of A'-inner automor- phisms [8]. Lemma Y If o G Aut(,4) is X-inner, say r" — s'xrs, all r G A for some s G Q()(A), then there exist nonzero a, b G A such that sa = b and arb = baraa, all r G A. Conversely if arb = b"r"a, all r G A, for some nonzero a, b G A, then there exists s G Q0( A ) with sa = b which induces o. Received bv the editors April 23, 19X2and. in revised form, luly 19, 19X2. 19X0Mathematics Subject Classification. Primary I6A72, 16A3X. ' Research supported in part by NSF grant No. MCS XI-01730. ' 19X3 American Mathematical Society 0OO2-9939/82/0OO0-O756/$02.75 569 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 570 SUSAN MONTGOMERY Now say that A is filtered; that is, A — UH,0A„, where the A„ are additive subgroups of A, 1 G A{), and AnAm G An + m, all n, m. For any a G A, let f(a) denote the filtration of a (that is, f(a) — n if a G An but a £ An_¡). We say that / is a degree function if f(ab) — f(a) + f(b), all a, b G A. The next lemma is essentially Proposition 1, part (1), of (I). Lemma 2. Let A be filtered with a degree function f. If a is an X-inner automorphism of A, thenf(a") = f(a). alla G A. We first consider the ring of generic matrices, and show that it has a degree function. For general facts about PI rings, we refer the reader to [3]. Let U = k{Xs,_Xm....} denote the ring of n X n generic matrices over an infinite field k. We may identify U with F/Tn, where F = k{y{.y„r...} is the free algebra and Tn is the T-ideal of F of all identities of nX n matrices. F is a graded algebra using the usual degree function d; that is, F — 2m>0 © Fm, where Fm is the /c-space spanned by the homogeneous monomials of degree m. If k is an infinte field, a standard argument shows that Tn is a graded ideal (if p — 2 p„, G Tn, where Pm G ^m<replace eachy, by Xy¡, 0 ¥= X G k, and use a Vandermonde determinant to see that pm G Tn). Thus U = F/T„ is also a graded algebra, where for any O^nGi/, the filtration of a is given by/(a) = min{d(p) \a = p + T„,p G F). Now U is a domain by the theorem of Amitsur (see [3, p. 90]). In a graded ring which is a domain, the filtration is additive, and so is a degree function. We summarize the above remarks as a lemma. Lemma 3. Let U — F/Tn be the ring of generic matrices over an infinite field k. Then U has an (additive) degree function which is induced from the degree function in F. Theorem 1. Let U = k(X) be the ring of generic matrices over an infinite field k. If o G Aul(U) fixes the center of U, then o = 1. Proof. By Posner's theorem, U has a quotient division ring £>; so D = UZ'\ where Z is the center of U, and C, the center of D, is just the quotient field of Z [3]. Thus o, when extended to D, fixes C, so is inner on D by the Skolem-Noether theorem. Since D — UZ~\ we may assume that a is induced by an element a £ U. Now for each generic matrix A",, aX° = X,a. Using the degree function in the lemma and the fact that f(X,) = 1, we see that /(A",") = 1 also. Thus X° = 1"'=, Oj + yl, some a , y G k. Since a ' ' G D, a is regular in U, and so a is also invertible considered as a matrix in Mn(k(x'tJ)), where the [x',j)l=i2 are the entires of the generic matrices (see [1, Lemma 10.3]). Thus we can view X° = a~xX,a as an equation in matrices; it follows that tr(A7) = tr( A",)and Det(A7) = DetiA",). Then tr(A",°)= 2™=,a]-tr( Xj ) + ny = tr(A",) implies that a, = 1 and a; = 0 if j # i, since 2"= i*// is algebraically independent over /c from 2"«t 2%ij+ixít- Moreover «y = 0. Thus Xf = X¡ + yl. But now, since Det(*7) = Det(A", + y) = Det( A",),it follows that y = 0. That is, A","= X„ for all /. We have shown that a = 1. As a consequence of Theorem 1, if G is any finite group of automorphisms of the generic matrix ring U, then the fixed ring UG has the same pi degree as U [5, Theorem 4.2]. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A'-INNER AUTOMORPHISMS. II 571 We now turn to differential polynomial rings. Let R be a prime ring with 1 with a derivation d, and let A — R[x; d], the differential polynomial ring in which xr = rx + d(r), all r G R. In the case when R is a commutative domain, the A'-inner automorphisms of A were described in (I), Theorem 2. Namely, let F he the quotient field of R, and D the quotient division ring of A (as A is an Ore domain in this case), and let a G Aul(A). If a becomes inner on D, say induced by s — b(x)a(x)~] G D, then o is also induced by q = ß„a~' G F, where ßn and am are the leading coeffi- cients of b(x) and a(x) respectively. In particular it follows that s = cq, where c is in the center of D. We show here that a similar result holds when the coefficient ring R is any domain; in fact something can be said when R is prime. Our argument is an extension of the proof in (I). Thus, consider an A'-inner automorphism o of A = R[x; d], for R prime. A is filtered, using An = {all polynomials of degree < n), and we let f(p) denote the "degree" of any p G A. If R is a domain, then /( pq) = f(p) + f(q), all p, q G A; thus by Lemma 2, o preserves the degree of any polynomial p G A. Now by Lemma 1, a is induced by some unit í G Q0(A), and there exist polynomials a = a(x), b = b(x) G A so that sa = b ¥= 0 and (*) apb = b°p"a, ah> G A. Write a = amxm + ■■ ■ +a0 and b = ß„x" + • • • +ß0, a„ ß, G R. Theorem 2. Let A = R[x; d], where R is prime with derivation d. Let o be an X-inner automorphism which preserves the degree of any polynomial. Let s G Q0( A ) induce a, and let a, b G A be as described above. Then a is also induced by q G Q0( R ), where qam — ßn ¥= 0 and ampßn ¥= ß°p"am, all p G A.

Load more

  7

Copy Link