Relative torsionfree modules with respect to a faithfully balanced bimodule

Abstract

Let S and R be two associative rings, be a faithfully balanced (S, R)-bimodule. We introduce and investigate properties of ∞-torsionfree modules with respect to which are generalizations of n-torsionfree modules for any positive integer n and totally C-reflexive modules. And we give some characterizations of the class of the ∞- C-torsionfree modules over any ring R. Moreover, when C is a semidualizing (S, R)-bimodule, we study the relations between the classes of relative torsionfree modules and the torsionfree modules in terms of Foxby equivalence. Finally, as the ∞-C-torsionfree module which is in the left orthogonal class of C is exactly the C-Gorenstein projective module, we characterize the Auslander class with respect to a semidualizing module by virtue of the ∞-C-torsionfree modules over the commutative Cohen-Macaulay ring with a dualizing module, which is a module version of Theorem 4.6 in Holm, H., Jørgensen, P. (2006). [Semi-dualizing modules and related Gorenstein homologica dimensions. J.Pure Appl. Algebra 205(2):423–445].