Adjoint functors, preradicals and closure operators in module categories

In this article preradicals and closure operators are studied in an adjoint situation, defined by two covariant functors between the module categories $R$-Mod and $S$-Mod. The mappings which determine the relationship between the classes of preradicals and the classes of closure operators of these categories are investigated. The goal of research is to elucidate the concordance (compatibility) of these mappings. For that some combinations of them, consisting of four mappings, are considered and the commutativity of corresponding diagrams (squares) is studied. The obtained results show the connection between considered mappings in adjoint situation.