Co-Artinian rings and Morita duality

Abstract

A ringA is left co-Noetherian if the injective hull of each simple leftA-module is Artinian. Such rings have been studied by Vamos and Jans. Dually, callA left co-Artinian if the injective hull of each simple leftA-module is Noetherian. Left co-Artinian rings having only finitely many nonisomorphic simple left modules are studied, and such rings are shown to have nilpotent radical. Moreover, it is shown that left co-Artinian implies left co-Noetherian ifA/J is Artinian. For an injective leftA-module A Q withB=End ( A Q), andC=End (Q B ), conditions yielding a Morita duality between\(\mathfrak{W}_B \) and\({}_C\mathfrak{W}\) are obtained. In special cases, e.g. A Q a self-cogenerator, this Morita duality yields chain conditions on A Q. Specialized to commutative rings, these results give the known fact that every commutative co-Artinian ring is co-Noetherian. Finally in the case that the injective hull A E=E( A S) of a simple leftA-module A S is a self-cogenerator, chain conditions on A E are related to chain conditions onB B =End ( A E). The results obtained are analogous to results for commutative rings of Vamos, Rosenberg and Zelinsky. It is shown that ifA is a left co-Artinian ring withE( A S) a self-cogenerator for each simple A S, thenJ is nil and\({}_Q\mathfrak{W}\).