Modules whose maximal submodules have τ-supplements

Abstract

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