Geodesic
Closure operators in modules and adjoint functors,~II
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2018), pp. 101-112.

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In this work we study the relations between the closure operators of two module categories connected by two adjoint contravariant functors. The present article is a continuation of the paper [1] (Part I), where the same question is investigated in the case of two adjoint covariant functors. An arbitrary bimodule $_RU_S$ defines a pair of adjoint contravariant functors $H_1=Hom_R(\text{-},U)\colon R\text{-}\mathrm{Mod}\to\mathrm{Mod}\text{-}S$ and $H_2=Hom_S(\text{-},U)\colon\mathrm{Mod}\text{-}S\to R\text{-}\mathrm{Mod}$ with two associated natural transformations and . In this situation we study the connections between the closure operators of the categories $R\text{-}\mathrm{Mod}$ and $\mathrm{Mod}\text{-}S$. 
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A. I. Kashu. Closure operators in modules and adjoint functors,~II. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2018), pp. 101-112. http://geodesic.mathdoc.fr/item/BASM_2018_2_a8/

[1] Kashu A. I., “Closure operators in modules and adjoint functors, I”, Algebra and Discrete Mathematics, 25:1 (2018), 98–117 | MR | Zbl

[2] Kashu A. I., Functors and torsions in module categories, Academy of Sciences of Moldova, Institute of Mathematics, Kishinev, 1997 (in Russian) | MR

[3] Kashu A. I., Radicals of modules and adjoint functors, Preprint, Academy of Sciences of MSSR, Institute of Mathematics, Kishinev, 1984 (in Russian)

[4] Dikranjan D., Tholen W., Categorical structure of closure operators, Kluwer Academic Publishers, 1995 | MR | Zbl

[5] Dikranjan D., Giuli E., “Factorizations, injectivity and compactness in categories of modules”, Commun. in Algebra, 19:1 (1991), 45–83 | DOI | MR | Zbl