Completions and classical localizations of right Noetherian rings

Abstract

Given a right Noetherian ring R and a prime ideal P of R, the injective hull of the right j?-module RIP is a finite power of a uniquely determined indecomposable injective I P . One forms the ring of right quotients R P of R relative to IP and the right ideal M = PR P of R P generated by P. The Jf-adic and Jp-adic topologies are compared; they turn out to coincide on every finitely generated .Rp-module when R P is a classical quasi-local ring with maximal ideal M. This condition also implies that R satisfies the right Ore condition with respect to the multiplicative set ^(P) introduced by Goldie, that the ikf-adic completion R P of R P is the bicommutator of IP, and that R P is an n by n matrix ring over a complete local ring.