A generalization of supplemented modules

Abstract

Let R be an arbitrary ring with identity and M a right R-module. In this paper, we introduce a class of modules which is an analogous of δ-supplemented modules defined by Kosan. The module M is called principally δ-supplemented, for all m∈M there exists a submodule A of M with M=mR+A and (mR)∩A δ-small in A. We prove that some results of δ-supplemented modules can be extended to principally δ-supplemented modules for this general settings. We supply some examples showing that there are principally δ-supplemented modules but not δ-supplemented. We also introduce principally δ-semiperfect modules as a generalization of δ-semiperfect modules and investigate their properties.