Some Remarks on Affine Rings

Various topics on affine rings are considered, such as the relationship between Gelfand-Kirillov dimension and Krull dimension, and when a "locally affine" algebra is affine. The dimension result is applied to study prime ideals in fixed rings of finite groups, and in identity components of group-graded rings. 0. Introduction. We study affine rings (i.e., rings finitely generated as algebras over a central subring) and group actions on them. Of particular interest is the relationship between the Gelfand-Kirillov (GK) dimension and the classical Krull dimension. Our first theorem shows for affine prime PI rings over a field that the GK dimension of a prime subalgebra is the same as its Krull dimension. This result is then applied to give a rapid proof of a result of Alev [1], on prime ideals in fixed rings. Later we turn to group-graded affine rings deriving results similar to those previously obtained for group actions; the notion of equivalence is introduced for prime ideals in the identity component, and an analog of Alev's theorem is proved. We also consider when a prime PI algebra which is "locally affine" must be affine. Some of the results in this paper (including Theorem 1) were announced at the NATO A.S.I, in Ring Theory, Antwerp, 1983 (see [7]). 1. Prime subrings of PI rings. We begin by stating a form of the Artin-Tate lemma, which will be used throughout this paper. Lemma 1. Let R c S be algebras over a commutative Noetherian ring C, such that S is a finite module over R and affine over C. If R is contained in the center of S, then R is C-affine. Consequently R is Noetherian and S is a Noetherian R-module. Proof. The usual commutative argument works in this situation; see also Lemma 27 of [11]. Our first theorem extends a result of Malliavin [4] on affine PI algebras to prime subrings of such rings, which are not necessarily affine. For any algebra A over a field k, we let c\(A) denote the classical Krull dimension of A, and GK(^4) denote the Gelfand-Kirillov dimension of A. Theorem 1. Let A be a prime PI algebra which is affine over a field k. If B is a prime subalgebra of A, then GK(5) = cl(5). Received by the editors March 25, 1985 and, in revised form, August 19, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 16A38,16A72. 1 Both authors acknowledge support from the NSF. ©1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page