Preradicals and closure operators in modules: comparative analysis and relations

Abstract

The theory of radicals in modules is based by the notion of preradical (as subfunctor of identical functor) [1]. The other important notion of the modern algebra is the closure operator (as a function C which by every pair of modules N ⊆ M defines a submodule CM(N) ⊆ M, C being compatible by the morphisms of R-Mod) [2]. The purpose of this communication consists in the elucidation of the relations between these fundamental notions and the comparison of results of those respective theories. The closure operators in some sense are the generalization of preradicals, since the class PR(R) can be inserted in CO(R) (by two methods). This important fact determines a close connection between the results of the respective domains.