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. © 2022 Elsevier B.V.