On a question of Mohamed and M\"uller

TL;DR

This paper investigates H-supplemented modules, explores their properties, and provides a negative answer to a question posed by Mohamed and Müller in 1990, advancing the understanding of module theory.

Contribution

It introduces and analyzes properties of H-supplemented modules and resolves a longstanding open question in module theory.

Findings

Direct summands and sums of H-supplemented modules are characterized.

The question by Mohamed and Müller from 1990 is answered negatively.

New insights into the structure of H-supplemented modules are provided.

Abstract

A module $M$ is called H-supplemented if for every submodule $A$ of $M$ there is a direct summand $A'$ of $M$ such that $A+X=M$ holds if and only if $A'+X=M$ for any submodule $X$ of $M$. (Equivalently, for each $X\leq M$, there exists a direct summand $D$ of $M$ such that $(X+D)/D\ll M/D$ and $(X+D)/X\ll M/X$.) Direct summands and sums of H-supplemented modules are studied and a question posed by Mohamed and M\"uller in 1990 is answered in the negative.