RD-flatness and RD-injectivity of simple modules

Abstract

We say that a ring R is a right RDV-ring if each simple right R -module is RD -injective. In this paper, we study the notion of RDV -rings which is a non-trivial generalization of V-rings and Köthe rings. For instance, commutative RD -rings, serial rings and right duo right uniserial rings are RDV -rings. Several characterizations of right RDV -rings are given. Also, it is shown that over a semilocal ring R with Jacobson radical J , each simple right R -module is RD -flat if and only if R is a left RDV -ring, if and only if R ( R / J ) is RD -injective, if and only if ( R / J ) R is RD -flat. As a consequence, we show that a local ring R is a principal ideal ring if and only if R satisfies the ascending chain condition on principal left ideals and R ( R / J ) is RD -injective. In the case of R being either a local left perfect ring or a normal left perfect ring, we have obtained results which state that to check whether every left R -module is RD -injective (or, R is Köthe), it suffices to test only the RD -injectivity of the simple left R -modules. Finally, we give some characterizations of quasi-Frobenius rings by using these concepts.