On a question of Mohamed and Müller

Abstract

A module $M$ is called H-supplemented if for every submodule $A$ of $M$ there is a direct summand $A^{\prime }$ of $M$ such that $A+X=M$ holds if and only if $A^{\prime }+X=M$ for any submodule $X$ of $M$. (Equivalently, for each $X\le M$, 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 Müller in 1990 is answered in the negative.