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