Generalization of reduction and closure of ideals

Document Type : Research Paper


1 Department of Mathematics, Faculty of Science, University of mohaghegh Ardabili, Ardabil

2 Department of Mathematics, University of Mohaghegh Ardabili, Ardabil, Iran



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.


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