Duality Pairs Induced by One-Sided Gorenstein Subcategories

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 R, S be rings and C a semidualizing (R, S)-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.