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 <