On a recent generalization of semiperfect rings

TL;DR

This paper corrects a recent claim about generalized supplemented rings, clarifies their relationship with semilocal and semiperfect rings, and introduces a broader concept of local submodules.

Contribution

It refutes Ding and Wang's claim, clarifies the hierarchy of ring classes, and proposes a new, wider notion of local submodules.

Findings

The class of generalized supplemented rings is between semilocal and semiperfect rings.

Ding and Wang's claim about generalized supplemented rings being semiperfect is false.

A new broader concept of local submodules is introduced.

Abstract

It follows from a recent paper by Ding and Wang that any ring which is generalized supplemented as left module over itself is semiperfect. The purpose of this note is to show that Ding and Wang's claim is not true and that the class of generalized supplemented rings lies properly between the class of semilocal and semiperfect rings. Moreover we rectify their claim by introducing a wider notion of local submodules.