فهرست نشریات علمی دسترسی آزاد ایران

Throughout this paper, all rings are commutative  with identity and all modules are unital. Let $R$  be a ring and $M$ be an $R$-module. Then $M$ is called a multiplication module provided for every submodule $N$ of $M$ there exists an ideal $I$ of $R$ such that $N=IM$. Also $M$ is said to be a comultiplication module if for every submodule $N$ of $M$ there exists an ideal $I$ of $R$ such that $ N=(0:_MI)$. In this  paper, we introduce the notions of reduction and coreduction of submodules, integral dependence, integral codependence, integral closure and $\Delta$-closure over multiplication and comultiplication modules.