Gorenstein transpose with respect to a semidualizing bimodule

Let S and R be rings and _SC_R be a semidualizing bimodule. We first give the definitions of C-transpose and n-C-torsionfree and give a criterion for a module A to be G_C-projective by some property of the C-transpose of A. Then we introduce the notion of C-Gorenstein transpose of a module over two-sided Noetherian rings. We prove that a module M in mod R^{op} is a C-Gorenstein transpose of a module A in mod S if and only if M can be embedded into a C-transpose of A with the cokernel G_C-projective. Finally we investigate some homological properties of the C-Gorenstein transpose of a given module.