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 = (0 :m i). in this paper, we introduce the notions of reduction and coreduction of submodules, integral dependence, integral codependence, integral closure and δ-closure over multiplication and comultiplication modules.