$H$-supplemented modules and singularity

Abstract

Let $M$ be a module over a ring $R$. We call $M$,$delta$-$H$-supplemented provided for every submodule $N$ of $M$ there is a direct summand $D$ of $M$ such that $M=N+X$ if and only if $M=D+X$ for every submodule $X$ of $M$ with $M/X$ singular. We prove that $M$ is $delta$-$H$-supplemented if and only if for every submodule $N$ of $M$ there exists a direct summand $D$ of $M$ such that $(N+D)/Nll_{delta} M/N$ and $(N+D)/Dll_{delta} M/D$.