Rad-supplementing modules

TL;DR

This paper introduces Rad-supplementing modules, characterizes their properties, and links their existence to ring conditions like left perfectness, providing new insights into module and ring theory.

Contribution

It defines Rad-supplementing modules, explores their properties, and establishes criteria relating them to ring structures such as left perfect rings.

Findings

Modules with composition series are Rad-supplementing.

A module has a Rad-supplement in its injective envelope iff in every essential extension.

R is left perfect iff R is semilocal, reduced, and certain sums are Rad-supplementing.

Abstract

Let R be an associative ring with unity and let M be an R-module. We call M (ample) Rad-supplementing if M has a (ample) Rad-supplement in every extension. If M is Rad-supplementing, then every direct summand of M is Rad-supplementing, but not each factor module of M. Every module with composition series is (Rad-)supplementing. M has a Rad-supplement in its injective envelope if and only if M has a Rad-supplement in every essential extension. R is left perfect if and only if R is semilocal and reduced, and the direct sum of countably many copies of R is Rad-supplementing if and only if R is reduced and the direct sum of countably many copies of R is ample Rad-supplementing. M is ample Rad-supplementing if and only if every submodule of M is Rad-supplementing. Every left R-module is (ample) Rad-supplementing if and only if R/P(R) is a left perfect ring, where P(R) is the sum of all left ideals of R such that Rad I=I.