Properties of content semimodules

Abstract

A semimodule M over semiring S is called a content semimodules if for every x ∈ M, then x ∈ c(x}M which c(x) = ∩{I|I ideal of S, x ∈ IM} and c is a function from M to I is called the content function. Semimodule is generalization of module, so the study of properties content modules that apply in content semimodules is needed. The aim of this study is investigated properties of semimodules that apply in content semimodules. Indeed we prove content semimodule M satisfy s(c(x)) = c(sx) for all x ∈ M if and only if (I: Ss}M = (IM: M s}. Furthermore, if M is content torsionfree semimodule over semidomain S then s(c(x)) = c(sx) for all s ∈ S and x ∈ M. By adopting a concept of semidomain, it has been proven that if every principal ideal of S is subtractive and M be a normally flat semimodules then M is torsionfree semimodule, implies that M is normally flat content semimodules so that applies s(c(x)) = c(sx). If M is content semimodule and I ideal of semidomain S that satisfy (I: S s}M = (IM: M s) then (I: S J}M = (IM: M J} for all I, J ideal of S, implies that for normally flat content semimodule M over semidomain S applies (I: SJ}M = (IM: MJ}.