$H$-supplemented modules and singularity

Document Type: Research Paper


Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar



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 $(N+D)/N\ll_{\delta} M/N$ and $(N+D)/D\ll_{\delta} M/D$.


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