Duality pairs induced by Auslander and Bass classes
Abstract Let R and S be arbitrary rings and let C S R {{}_{R}C_{S}} be a semidualizing bimodule, and let 𝒜 C ( R op ) {\mathcal{A}_{C}(R^{\mathrm{op}})} and ℬ C ( R ) {\mathcal{B}_{C}(R)} be the Auslander and Bass classes, respectively. Then both pairs ( 𝒜 C ( R op ) , ℬ C ( R ) ) and ( ℬ C ( R ) , 𝒜 C ( R op ) ) (\mathcal{A}_{C}(R^{\mathrm{op}}),\mathcal{B}_{C}(R))\quad\text{and}\quad(% \mathcal{B}_{C}(R),\mathcal{A}_{C}(R^{\mathrm{op}})) are coproduct-closed and product-closed duality pairs and both 𝒜 C ( R op ) {\mathcal{A}_{C}(R^{\mathrm{op}})} and ℬ C ( R ) {\mathcal{B}_{C}(R)} are covering and preenveloping; in particular, the former duality pair is perfect. Moreover, if ℬ C ( R ) {\mathcal{B}_{C}(R)} is enveloping in Mod R {\operatorname{Mod}R} , then 𝒜 C ( S ) {\mathcal{A}_{C}(S)} is enveloping in Mod S {\operatorname{Mod}S} . Also, some applications to the Auslander projective dimension of modules are given.