Closure operators in the categories of modules Part I (Weakly hereditary and idempotent operators)

Abstract

In this work the closure operators of a category of modules \(R\)-Mod are studied. Every closure operator \(C\) of \(R\)-Mod defines two functions \( \mathcal{F}_1^{C}\) and \(\mathcal{F}_2^{C}\), which  in every module \(M\) distinguish the set of \(C\)-dense submodules  \(\mathcal{F}_1^{C}(M)\) and the set of \(C\)-closed submodules \(\mathcal{F}_2^{C}(M)\). By means of these functions three types of closure operators are described: 1)weakly hereditary; 2)idempotent; 3)weakly hereditary and idempotent.