$X$-inner automorphisms of filtered algebras

Abstract

We continue earlier work and compute the X-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 X-inner automorphisms of filtered algebras begun in [7], which will be referred to as (I) in what follows. In (I) we described the X-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 X-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 X-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 X-inner automorphisms is abelian. In what follows, A will always denote a prime ring. Recall that a E Aut(A) is X-inner if it becomes inner when extended to the (right) Martindale quotient ring QO(A) of A; when A is an Ore domain, a is X-inner if and only if it becomes inner on the quotient division ring D of A. We shall need the following properties of QO(A): it is the (right) quotient ring of A with respect to the filter IF of all nonzero two-sided ideals (that is, QO(A) = lim ,HomA('A, 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 QO(A) as left multiplications. By construction, for any 0 #d x E A, there exists a nonzero I of A so that 0 =# xI c A. Fundamental in what follows is an internal characterization of X-inner automorphisms [8]. LEMMA 1. If a E Aut(A) is X-inner, say r' = s-'rs, all r E A for some s E QO(A), then there exist nonzero 'a, b E A such that sa = b and arb = b0r0a, all r E A. Conversely if arb = b0r0a, 'all r E A, for some nonzero a, b E A, then there exists s E QO(A) with sa b which induces a. Received by the editors April 23, 1982 and, in revised form, July 19, 1982. 1980 Mathematics Subject Classification. Primary 16A72, 16A38. ' Research supported in part by NSF grant No. MCS 81-01730. ? 1983 American Mathematical Society 0002-9939/82/0000-0756/$02.75