A note on injective rings

Introduction.Injective modules, specially self-injective rings, occupy a prominent position in ring theory and have drawn the attention of many authors since several years (cf.for example the bibliography of [1], [3], [4], [6] ) .Well-known examples of self-injective rings are self-injective regular rings, quasi-Frobeniusean rings and pseud0-Frobeniusean rings.The purpose of this note is to consider several nice conditions for rings to be self-injective.Test modules are given to ensure that rings are left self -injective regular with non-zero socle.Sufficient conditions for rings to be pseud0-Frobeniusean and quasi-Frobeniusean follow.Strongly regular rings with non-zero socle are characterized.The following are among the results proved for a ring A:(1) If A contains an injective maximal left ideal Y such that r(Y) is a minimal right ideal, then A is left self-in- jective; (2) A is left self-injective if A is weakly right duo containing an injective maximal left ideal; (3) A is left pseud0-Frobeniusean if A is left p -injective left Kasch containing an injective maximal left ideal.Throughout, A denotes an asscoiative ring with identity and A-mod-