Krull dimension of modules over involution rings. II

Abstract

The following question of Lanski is answered positively in the case when a ring R with involution * is Noetherian with respect to two-sided *ideals. Let R be a ring with * and invertible 2, and let S be the subring of R generated by the symmetric elements in R. Does any left R-module have the same Krull dimension when considered as an R-module and S-module? Our starting point is a question implicit in a paper of Lanski [7], namely given a ring R with involution *, with 1/2 E R, and with S the subring generated by the symmetric elements in R, and given any left R-module M, does M have the same Krull dimension when considered as an R-module and as an S-module? The question is a very natural one, given developments in the theory of rings with involution in the 1970s, but is surprisingly hard to settle. Indeed Lanski [7, p. 398] frankly admits that he did not know the answer even in the case of an Artinian R-module M. To the best of our knowledge, this is still the case. Let us review some of the positive results related to this question. In an earlier paper, Lanski [5] proved that if R is an Artinian ring then so too is S and R is a finitely generated S-module. In [6], Lanski continued his investigations and proved that if R is a Noetherian ring then R is a Noetherian S-module and the Krull dimensions of R as an R-module and as an S-module are the same. The first part of this theorem of Lanski was proved independently by Chuang and Lee [1]. In this note we shall prove that if R satisfies the ascending chain condition on two-sided *-ideals then any left R-module has the same (dual) Krull dimension when viewed as an R-module and as an S-module. This theorem contains all the above results as special cases. As one might expect, the arguments in the cases of Krull dimension and Krull dimension are to each other in many places, but Lemma 7 and its dual Lemma 7* do differ in some important regards. To be careful and for the convenience of the reader, we give both arguments, the argument being given in parentheses. Naturally we are disappointed not to be able to settle the general question one way or the other, but we present our results in the belief that the methods are interesting. Basically, in the spirit of Lanski's work, we make certain reductions Received by the editors July 29, 1992 and, in revised form, September 29, 1992. 1991 Mathematics Subject Classification. Primary 16W10, 16P60. ? 1994 American Mathematical Society 0002-9939/94 $1.00 + $.25 per page