Cofinitely semiperfect modules

Abstract

It is well known that a projective module M is ⊕-supplemented if and only if M is semiperfect. We show that a projective module M is ⊕-cofinitely supplemented if and only if M is cofinitely semiperfect or briefly cof-semiperfect (i.e., each finitely generated factor module of M has a projective cover). In this paper we give various properties of the cof-semiperfect modules. If a projective module M is semiperfect then every M -generated module is cof-semiperfect. A ring R is semiperfect if and only if every free R-module is cof-semiperfect.

A Note on⊕-Cofinitely Supplemented Modules

“…In Section 3, we will show that an arbitrary module is cofinitely semiperfect if and only if it is (amply) cofinitely supplemented by supplements which have projective covers. This extends [1,Theorem 2.3].…”

“…M is called a cofinitely semiperfect module if every finitely generated factor module of M has a projective cover. The following result generalizes [1,Theorem 2.3].…”

“…M is called semiperfect if every factor module of M has a projective cover. Calisici and Pancar [1] introduced the notion of cofinitely semiperfect modules as a proper generalization of semiperfect modules. M is called a cofinitely semiperfect module if every finitely generated factor module of M has a projective cover.…”

“…A module M is called an ⊕-cofinitely supplemented module if every cofinite submodule of M has a supplement that is a direct summand of 2 International Journal of Mathematics and Mathematical Sciences M. In Section 2, we show that a quotient of an ⊕-cofinitely supplemented module is not in general ⊕-cofinitely supplemented. It is proven that if a module M is an ⊕-cofinitely supplemented multiplication module with Rad(M) M, then M can be written as an irredundant sum of local direct summand of M. Calisici and Pancar [1] defined the concept of cofnitely semiperfect modules and proved that for a projective module M, M is cofinitely semiperfect if and only if M is amply cofinitely supplemented. In Section 3, we will show that an arbitrary module is cofinitely semiperfect if and only if it is (amply) cofinitely supplemented by supplements which have projective covers.…”