\(F\)-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 \( F \subseteq X \subseteq M\) such that \( U + X = M\), or equivalently, \(F\subseteq V\), \(U + V = M\) and \(U \cap V \) 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.