$F$-supplemented modules

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.