S-small and S-essential submodules

This paper is concerned with S-comultiplication modules which are a generalization of comultiplication 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≠0 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≪SM, then there exist an ideal I≤SeR and an t∈S such that t(0:MI)≤N≤(0:MI). The converse is true if S⊆U(R) such that 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≤SeM if and only if there exist an ideal I≪SR and an s∈S such that s(0:MI)≤N≤(0:MI). In section 3, 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.