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Duality Pairs Induced by One-Sided Gorenstein Subcategories

  • Weiling Song
  • Tiwei Zhao
  • Zhaoyong HuangEmail author
Article
  • 19 Downloads

Abstract

For a ring R and an additive subcategory \(\mathscr {C}\) of the category \(\mathop {\mathrm{Mod}}\nolimits R\) of left R-modules, under some conditions, we prove that the right Gorenstein subcategory of \(\mathop {\mathrm{Mod}}\nolimits R\) and the left Gorenstein subcategory of \(\mathop {\mathrm{Mod}}\nolimits R^{op}\) relative to \(\mathscr {C}\) form a coproduct-closed duality pair. Let RS be rings and C a semidualizing (RS)-bimodule. As applications of the above result, we get that if S is right coherent and C is faithfully semidualizing, then \((\mathcal {GF}_C(R),\mathcal {GI}_C(R^{op}))\) is a coproduct-closed duality pair and \(\mathcal {GF}_C(R)\) is covering in \(\mathop {\mathrm{Mod}}\nolimits R\), where \(\mathcal {G}\mathcal {F}_C(R)\) is the subcategory of \(\mathop {\mathrm{Mod}}\nolimits R\) consisting of C-Gorenstein flat modules and \(\mathcal {G}\mathcal {I}_C(R^{op})\) is the subcategory of \(\mathop {\mathrm{Mod}}\nolimits R^{op}\) consisting of C-Gorenstein injective modules; we also get that if S is right coherent, then \((\mathcal {A}_C(R^{op}),l\mathcal {G}(\mathcal {F}_C(R)))\) is a coproduct-closed and product-closed duality pair and \(\mathcal {A}_C(R^{op})\) is covering and preenveloping in \(\mathop {\mathrm{Mod}}\nolimits R^{op}\), where \(\mathcal {A}_C(R^{op})\) is the Auslander class in \(\mathop {\mathrm{Mod}}\nolimits R^{op}\) and \(l\mathcal {G}(\mathcal {F}_C(R))\) is the left Gorenstein subcategory of \(\mathop {\mathrm{Mod}}\nolimits R\) relative to C-flat modules.

Keywords

Duality pairs Right Gorenstein subcategories Left Gorenstein subcategories C-Gorenstein flat modules C-Gorenstein injective modules Auslander classes Cotorsion pairs 

Mathematics Subject Classification

18G25 16E30 

Notes

Acknowledgements

This research was partially supported by NSFC (Grant No. 11571164) and the NSF of Shandong Province (Grant No. ZR2019QA015). The authors thank the referees for the useful suggestions.

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Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2019

Authors and Affiliations

  1. 1.Department of Applied Mathematics, College of ScienceNanjing Forestry UniversityNanjingPeople’s Republic of China
  2. 2.School of Mathematical SciencesQufu Normal UniversityQufuPeople’s Republic of China
  3. 3.Department of MathematicsNanjing UniversityNanjingPeople’s Republic of China

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