F-supplemented modules

Abstract

Let R be a ring, let M be a left R-module, and let U,V,F be submodules of M with F proper. We call V an F-supplement of U in M if V is minimal in the set F⊆X⊆M such that U+X=M, or equivalently, F⊆V, U+V=M and U∩V is F-small in V. If every submodule of M has an F-supplement, then we call M an F-supplemented module. In this paper, we introduce and investigate F-supplement submodules and (amply) F-supplemented modules. We give some properties of these modules, and characterize finitely generated (amply) F-supplemented modules in terms of their certain submodules.