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 2 M, there exists a decomposition M ¼ A È B such that A is projective, A Rx and Rx \ B 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 ZðMÞ; SocðMÞ; dðMÞ. We prove, among others, that if M is a finitely generated projective module, then M is quasi-injective if and only if M is ZðMÞ-semiregular and M È M is CS. If M is projective SocðMÞ-semiregular module, then M is semiregular. We also characterize QF-rings R with JðRÞ 2 ¼ 0.

A generalization of projective covers

“…[9,Theorem 2.4] or [1,Theorem 3.2]. Hence if R is Z( R R)-semiregular, then every finitely presented left R-module has a projective Z( R R)-cover by Theorem 3.11.…”

“…An element m of a module M is called I -semiregular if it satisfies the conditions in Lemma 2.1, and M is called an I -semiregular module if every element of M is I -semiregular. In [1], it is named by "I M-semiregular" but we use "I -semiregular" in this note for short. A ring R is called I -semiregular if R R is an I -semiregular module.…”

“…A module is called semiregular if each of its elements is semiregular. For a module M, this concept was extended in [1] to the notion of U -semiregular elements by taking any fully invariant submodule U of M instead of Rad(M).…”

“…Recently some authors have worked with various extensions of these rings (see for example [1,9,11,12,14,15]). Zhou calls a ring R δ-semiperfect (δ-semiregular) if for a (finitely generated) left ideal I of R, I = Rf ⊕ S where f 2 = f ∈ R and S ⊆ δ( R R), and also by defining projective δ-cover, he characterizes these rings.…”

“…Next, Nicholson and Zhou study I -semiperfect and I -semiregular rings by introducing strongly lifting ideals. A module theoretic version of these extensions is studied in [1,12]; also by defining projective Soc-covers, Soc-semiperfect modules are characterized in [12].…”