Modules whose maximal submodules have $\tau$-supplements

Abstract

Let $R$ be a ring and $\tau$ be a preradical for  the category of left $R$-modules. In this paper, we study on modules whose maximal submodules have $\tau$-supplements. We give some characterizations of these modules in terms their certain submodules, so called $\tau$-local submodules. For some certain preradicals $\tau$, i.e.  $\tau=\delta$ and idempotent $\tau$,  we prove that every maximal submodule of $M$ has a $\tau$-supplement if and only if every cofinite submodule of $M$ has a $\tau$-supplement. For a radical $\tau$ on $\operatorname{R-Mod}$, we  prove that, for every $R$-module every submodule is a $\tau$-supplement if and only if $R/\tau(R)$ is semisimple and $\tau$ is hereditary.