Quasi-Dual Modules

Let R be a ring, M be a right R-module and S = EndR(M). M is called a quasi-dual module if, for every R-submodule N of M, N is a direct summand of rM(X) where X \subseteq S. In this article, we study and provide several characterizations of this module classes. We show that if M is quasi-dual module, then, for all m \in M, rM \ellS(m) = mR \oplus K for some submodule K of M. We also show that every quasi-dual module is a Kasch module and Z(SM) \subseteq Rad (MR).