• Journal of the Korean Mathematical Society
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• v.55 no.6
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• pp.1389-1421
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• 2018

In this paper, we introduce the notion of C-Gorenstein weak injective modules with respect to a semidualizing bimodule $_SC_R$, where R and S are arbitrary associative rings. We show that an iteration of the procedure used to define $G_C$-weak injective modules yields exactly the $G_C$-weak injective modules, and then give the Foxby equivalence in this setting analogous to that of C-Gorenstein injective modules over commutative Noetherian rings. Finally, some applications are given, including weak co-Auslander-Buchweitz context, model structure and dual pair induced by $G_C$-weak injective modules.

• Journal of the Korean Mathematical Society
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• v.61 no.1
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• pp.29-40
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• 2024

Let S and R be rings and _SC_R a semidualizing bimodule. We introduce the notion of G_C-FP_n-injective modules, which generalizes G_C-FP-injective modules and G_C-weak injective modules. The homological properties and the stability of G_C-FP_n-injective modules are investigated. When S is a left n-coherent ring, several nice properties and new Foxby equivalences relative to G_C-FP_n-injective modules are given.

• Journal of the Korean Mathematical Society
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• v.54 no.5
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• pp.1457-1482
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• 2017

Let S and R be rings and $_SC_R$ a (faithfully) semidualizing bimodule. We introduce and study C-weak flat and C-weak injective modules as a generalization of C-flat and C-injective modules ([21]) respectively, and use them to provide additional information concerning the important Foxby equivalence between the subclasses of the Auslander class ${\mathcal{A}}_C$ (R) and that of the Bass class ${\mathcal{B}}_C$ (S). Then we study the stability of Auslander and Bass classes, which enables us to give some alternative characterizations of the modules in ${\mathcal{A}}_C$ (R) and ${\mathcal{B}}_C$ (S). Finally we consider an open question which is closely relative to the main results ([11]), and discuss the relationship between the Bass class ${\mathcal{B}}_C$(S) and the class of Gorenstein injective modules.

• Journal of the Korean Mathematical Society
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• v.53 no.1
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• pp.1-17
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• 2016

Let $\mathcal{W}$ be an additive full subcategory of the category R-Mod of left R-modules. We provide a method to construct a proper ${\mathcal{W}}^H_C$-resolution (resp. coproper ${\mathcal{W}}^T_C$-coresolution) of one term in a short exact sequence in R-Mod from those of the other two terms. By using these constructions, we introduce and study the stability of the Gorenstein categories ${\mathcal{G}}_C({\mathcal{W}}{\mathcal{W}}^T_C)$ and ${\mathcal{G}}_C({\mathcal{W}}^H_C{\mathcal{W}})$ with respect to a semidualizing bimodule C, and investigate the 2-out-of-3 property of these categories of a short exact sequence by using these constructions. Also we prove how they are related to the Gorenstein categories ${\mathcal{G}}((R{\ltimes}C){\otimes}_R{\mathcal{W}})_C$ and ${\mathcal{G}}(Hom_R(R{\ltimes}C,{\mathcal{W}}))_C$ over $R{\ltimes}C$.