When $\delta$-semiperfect rings are semiperfect

TL;DR

This paper explores the relationship between $ ext{delta}$-semiperfect rings and semiperfect rings, establishing that semiperfect rings are exactly the semilocal $ ext{delta}$-supplemented rings, with module-theoretic implications.

Contribution

It characterizes semiperfect rings in terms of $ ext{delta}$-supplemented semilocal rings and extends these concepts to module theory.

Findings

Semiperfect rings are precisely the semilocal $ ext{delta}$-supplemented rings.

Provides module-theoretic versions of the ring-theoretic results.

Establishes a characterization linking $ ext{delta}$-semiperfect and semiperfect rings.

Abstract

Zhou defined $\delta$-semiperfect rings as a proper generalization of semiperfect rings. The purpose of this paper is to discuss relative notions of supplemented modules and to show that the semiperfect rings are precisely the semilocal rings which are $\delta$-supplemented. Module theoretic version of our results are obtained.