Totally reflexive modules with respect to a semidualizing bimodule

Let S and {iaR} be two associative rings, let _S C _R be a semidualizing (S,R)-bimodule. We introduce and investigate properties of the totally reflexive module with respect to _S C _R and we give a characterization of the class of the totally C _R -reflexive modules over any ring R. Moreover, we show that the totally C _R -reflexive module with finite projective dimension is exactly the finitely generated projective right R-module. We then study the relations between the class of totally reflexive modules and the Bass class with respect to a semidualizing bimodule. The paper contains several results which are new in the commutative Noetherian setting.