A generalization of the Auslander transpose and the generalized Gorenstein dimension

Let R be a left and right Noetherian ring and C a semidualizing R-bimodule. We introduce a transpose Tr _c M of an R-module M with respect to C which unifies the Auslander transpose and Huang’s transpose, see Z.Y.Huang, On a generalization of the Auslander-Bridger transpose, Comm. Algebra 27 (1999), 5791–5812, in the two-sided Noetherian setting, and use Tr _c M to develop further the generalized Gorenstein dimension with respect to C. Especially, we generalize the Auslander-Bridger formula to the generalized Gorenstein dimension case. These results extend the corresponding ones on the Gorenstein dimension obtained by Auslander in M. Auslander, M. Bridger, Stable Module Theory, Mem. Amer. Math. Soc. vol. 94, Amer. Math. Soc., Providence, RI, 1969.