Closure operators in modules and adjoint functors, I

Abstract

In the present work the relations between the closure operators of two module categories are investigated in the case when the given categories are connected by two covariant adjoint functors \(H\colon R\mbox{-Mod}\longrightarrow  S\)-Mod and   \(T\colon S\mbox{-Mod} \longrightarrow R\)-Mod. Two mappings are defined which ensure the transition between the closure operators of categories \(R\)-Mod and \(S\)-Mod. Some important properties of these mappings are proved. It is shown that the studied mappings are compatible with the order relations and with the main  operations.