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 F₁с and F₂с, which in every module M distinguish the set of C-dense submodules F₁с(M) and the set of C-closed submodules F₂с(M). By means of these functions three types of closure operators are described: 1) weakly hereditary; 2) idempotent; 3) weakly hereditary and idempotent.