Continuous rings with chain conditions

Abstract

The quasi-Frobenius rings are characterized as the left continuous rings satisfying either (A 1) or (A 2) and either (S 1) or (S 2), where these conditions are defined as follows: (A 1): ACC on left annihilators; ( A 2): R/ Soc( R R) is left Goldie; ( S 1): S = r( l( S)) for every minimal right ideal S; and (S 2): Every minimal right ideal is essential in a summand of R R . These characterizations extend several results in the literature. In addition, it is shown that, in these rings, Soc( R R ) = Soc( R R ), Soc( eR) is simple for every primitive idempotent e of R, and there exists a complete set of distinct representatives { Rt 1,..., Rt n } of the isomorphism classes of the simple left R-modules such that { t 1 R,..., t n R} is a complete set of distinct representatives of the isomorphism classes of the simple right R-modules.