Characterizations of I-semiregular and I-semiperfect rings

TL;DR

This paper introduces and characterizes I-semiregular and I-semiperfect rings using (locally)projective I-covers and I-supplemented modules, generalizing existing concepts and providing new characterizations.

Contribution

It defines (locally)projective I-covers and uses them to characterize I-semiregular and I-semiperfect rings, extending prior work with new module-theoretic approaches.

Findings

Characterization of I-semiregular rings via projectivity classes.

Characterization of I-semiperfect rings using I-covers.

Introduction of I-supplemented modules and their role in ring characterization.

Abstract

Let $M$ be a left module over a ring $R$ and $I$ an ideal of $R$. We call $(P, f)$ a (locally)projective $I$-cover of $M$ if $f$ is an epimorphism from $P$ to $M$, $P$ is (locally)projective, $Kerf\subseteq IP$, and whenever $P=Kerf+X$, then there is a projective summand $Y$ of $P$ in $Kerf$ such that $P=Y\oplus X$. This definition generalizes (locally)projective covers. We characterize $I$-semiregular and $I$-semiperfect rings which are defined by Yousif and Zhou [19] using (locally)projective $I$-covers in section 2 and 3. $I$-semiregular and $I$-semiperfect rings are characterized by projectivity classes in section 4. Finally, the notion of $I$-supplemented modules are introduced and $I$-semiregular and $I$-semiperfect rings are characterized by $I$-supplemented modules. Some well known results are obtained as corollaries.