Localization in a Morita context with applications to fixed rings

Abstract

In the study of a noetherian ring R, there are two related questions that have played a major role in recent years; one is finding prime ideals of R which satisfy the second layer condition, the other is finding the (second layer) link closed sets of prime ideals of R. In particular these properties are necessary to determine the classical sets of prime ideals of R, which allows one to obtain a well behaved Ore localization (see, for example, [24, 8, 91). One aspect of this problem is the transfer of these properties from the prime spectrum of one ring R to the prime spectrum of another, related ring, A. Jategaonkar [S] has shown that, if R is a commutative noetherian ring, then the group ring RG satisfies the second layer condition whenever G is polycyclic-by-finite. This result was then generalized by Bell [2] to strongly group graded rings. More recently Letzter [lo] has studied the relationship between the prime spectrums of the noetherian rings R and S, when S is module finite over R, showing how the second layer condition and links between prime ideals transfer from one ring to the other. Let T be a right noetherian ring and let e be an idempotent element of T, then it is well known that eTe is a right noetherian ring. In Sections 1 and 2 we examine the transfer of the second layer condition between the prime ideals of T and eTe, as well as the transfer of classical sets of prime ideals between these rings. This enables us to study how these properties of Spec(eTe) translate to Spec(( 1 e) T( 1 e)). In particular, under certain hypotheses, we prove that whenever eTe satisfies the second layer condition, then so does (1 e) T( 1 e) (Corollary 2.6). In addition we are able to show that under the same hypotheses, if every prime ideal of eTe belongs to a finite classical set, then so does every prime ideal of (1 -e) T(l -e). Another way to describe the relationship between T and eTe is via a Morita context (see [ 11). Let R and S be rings, let s W, and R V, be bimodules, and let 6: VOs W-+ R, $: WOR V+ S be bimodule 373 OOZI-8693/91 $3.00