Closure operators in modules and adjoint functors, I

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 $H\colon R\text{-}\operatorname{Mod}\longrightarrow S\text{-}\operatorname{Mod}$ and $T\colon S\text{-}\operatorname{Mod} \longrightarrow R\text{-}\operatorname{Mod}$. Two mappings are defined which ensure the transition between the closure operators of categories $R\text{-}\operatorname{Mod}$ and $S\text{-}\operatorname{Mod}$. 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.