Semiregular Modules with Respect to a Fully Invariant Submodule

Abstract

Let M be a left R-module and F a submodule of M for any ring R. We call M F-semiregular if for every x is an element of M, there exists a decomposition M = A circle plus B such that A is projective, A less than or equal to Rx and Rx boolean AND B less than or equal to F. This definition extends several notions in the literature. We investigate some equivalent conditions to F-semiregular modules and consider some certain fully invariant submodules such as Z(M), Soc(M), delta(M). We prove, among others, that if M is a finitely generated projective module, then M is quasi-injective if and only if M is Z(M)-semiregular and M circle plus M is CS. If M is projective Soc(M)-semiregular module, then M is semiregular. We also characterize QF-rings R with J(R)(2) = 0.