S-SMALL AND S-ESSENTIAL SUBMODULES

This paper is concerned with S-comultiplication mod ules 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 S M, then there exist an ideal I ≤S e R and an t ∈ S such that t(0 :M I) ≤ N ≤ (0 :M I). 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 ≤S e M if and only if there exist an ideal I s R and an s ∈ S such that s(0 :M I) ≤ N ≤ (0 :M I). 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.