Vol. 48, No. 1, 1973

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Completions and classical localizations of right Noetherian rings

Joachim Lambek and Gerhard O. Michler

Vol. 48 (1973), No. 1, 133–140

Given a right Noetherian ring R and a prime ideal P of R, the injective hull of the right R-module R∕P is a finite power of a uniquely determined indecomposable injective IP. One forms the ring of right quotients RP of R relative to IP and the right ideal M = PRP of RP generated by P. The M-adic and IP-adic topologies are compared; they turn out to coincide on every finitely generated RP-module when RP 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 𝒞(P) introduced by Goldie, that the M-adic completion RP of RP is the bicommutator of IP, and that RP is an n by n matrix ring over a complete local ring.

Mathematical Subject Classification
Primary: 16A08
Received: 7 June 1972
Published: 1 September 1973
Joachim Lambek
Gerhard O. Michler