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.