S-small and S-essential submodules

This paper is concerned with S-comultiplication modules which are a generalization of comultiplication modules.In section ۲, we introduce the S-small and S-essential submodules of a unitary R-module M over a commutative ring R with ۱\neq ۰ such that S is a multiplicatively closed subset of R. We prove that if M is a faithful S-strong comultiplication R-module and N\ll ^{S}M, then there exist an ideal I\leq ^{S}_{e}R and an t\in S such that t(۰ :_{M}I)\leq N\leq (۰ :_{M}I). The converse is true if S\subseteq {\rm U}(R) such that {\rm U}(R) is the set of all units of R. Also, we prove that if M is a torsion-free S-strong comultiplication module, then N\leq ^{S}_{e} M if and only if there exist an ideal I\ll ^{S}R and an s\in S such that s(۰ :_{M} I)\leq N\leq (۰ :_{M} I). In section ۳, we introduce the concept of S-quasi-copure submodule N of an R-module M and investigate some results related to this class of submodules.

Keywords:

‎‎‎S-‎small submodule , S-essential submodule , S-co-m module