A generalization of supplemented modules

TL;DR

This paper introduces a generalized concept of supplemented modules called I-supplemented modules, and uses this framework to characterize certain classes of rings, extending previous results in module and ring theory.

Contribution

It generalizes supplemented modules to I-supplemented modules and characterizes I-semiregular, I-semiperfect, and I-perfect rings using this new concept.

Findings

Characterization of I-semiregular rings

Characterization of I-semiperfect rings

Characterization of I-perfect rings

Abstract

Let $M$ be a left module over a ring $R$ and $I$ an ideal of $R$. $M$ is called an $I$-supplemented module (finitely $I$-supplemented module) if for every submodule (finitely generated submodule) $X$ of $M$, there is a submodule $Y$ of $M$ such that $X+Y=M$, $X\cap Y\subseteq IY$ and $X\cap Y$ is PSD in $Y$. This definition generalizes supplemented modules and $\delta$-supplemented modules. We characterize $I$-semiregular, $I$-semiperfect and $I$-perfect rings which are defined by Yousif and Zhou [15] using $I$-supplemented modules. Some well known results are obtained as corollaries.