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## Closure operators in modules and adjoint functors, II

Authors: Caşu Alexei

### Abstract

In this work we study the relations between the closure operators of two module categories connected by two adjoint \emph{contravariant} functors. The present article is a continuation of the paper \cite{kas1} (Part~I), where the same question is investigated in the case of two adjoint covariant functors.\\ \noindent\hspace*{6mm}An arbitrary bimodule $_{\ind R}U_{\!\ind S}$ defines a pair of adjoint contravariant functors $H_1 = Hom_{\ind R}(\mbox{-},U) : R\mbox{-Mod} \to \mbox{Mod-}S$ and $H_2 = Hom_{\ind S}(\mbox{-},U) : \mbox{Mod-}S \to R\mbox{-Mod}$ with two associated natural transformations $\Phi : \unu_{\ind {R\mbox{-\tiny Mod}}}\to H_2H_1$ \ and \ $\Psi : \unu_{\ind {\mbox{\tiny Mod-}S}}\to H_1H_2$. In this situation we study the connections between the closure operators of the categories $R\mbox{-Mod}$ and \ $\mbox{Mod-}S$.

Institute of Mathematics and Computer Science