S-small and S-essential submodules

This paper is concerned with S-co-m modules which are a generalization of co-m modules. In section 2, we introduce the S-small and S-essential submodules of a unitary $R$-module $M$ over a commutative ring $R$ with $1\neq 0$ such that S is a multiplicatively closed subset of $R$. We prove that if $M$ is an S-co-m module satisfying the S-DAC and $N\leq M$, then $N\leq ^{S}_{e}M$ if and only if there exists $I\ll ^{S}R$ such that $s(0:_{M}I)\leq N\leq (0 :_{M}I)$ for some $s\in S$. Let $M$ be a faithful S-strong co-m $R$-module. We prove that if $N\ll ^{S}M$ then there exists an ideal $I\leq ^{S}_{e}R$ such that $s(0 :_{M}I)\leq N\leq (0 :_{M}I)$. The converse is true if $S=\{1\}$and $M$ is a prime module. In section 3, we introduce the S-quasi-copure submodules $N$ of an $R$-module $M$ and investigate some results related to this class of submodules.