On noncommutative localization

Abstract

For a prime ideal P of a Noetherian ring R, Goldic defined in 17, 81 a localization Q of R at P. This localization is constructed as a subring of the inverse limit of the rings Q,,(R/P’“‘), where P(“) is the nth symbolic power of P. Note that R/P’“’ has an Artinian classical ring of quotients, which is denoted by Q,.,(R/P’“‘). The ring Q has been shown by Goldie to have the property that Q/J(Q) z Q,,(R/P) and fl?=, J(Q)’ = (0), where J(Q) is the Jacobson radical of Q. The inversive localization Rr,,, of R at P, introduced by Cohn in (61, is defined as follows. If r(P) denotes the set of matrices regular modulo P, then R T(,,j is the ring universal with respect to the property that all matrices in r(P) are invertible over R,.o,. It has been shown in [ 11 that RrcJ,, is universal with respect to the property that R I ,,,/J(R,.,,,) is naturally isomorphic to Q,,(R/P), and so there is a canonical mapping from R,-(,,, into Goldie’s localization Q. Theorem 2.4 shows that Q z R,,,,/T, where T is the intersection of powers of J(Rrc,,). Theorem 2.6 then characterizes Q as the ring universal with respect to the property that Q/J(Q) is naturally isomorphic to Q,,(R/P) and nz-, J(Q)” = (0). The results relating the two localizations depend on Theorem 1.4, which shows that if R is a left Noetherian ring with prime radical N, then R,(,,, is a left Artinian ring. In fact, R,.,,v, z Q,,(R/K), where K is the smallest ideal of R such that every element regular modulo N is also regular modulo K. This implies that R,(,, is left Artinian when P is a minimal prime ideal, and makes it possible to define the symbolic powers of any semiprime ideal. Comments at the end of the paper indicate how these results can be extended to rings with finite reduced rank (in the sense of 121). and thus the results hold for all rings with Krull dimension.