Generalization of reduction and closure of ideals

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=(۰:_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.

کلیدواژه ها:

Reduction ، Integral closure ، Multiplication modules ، integrally dependent