Remarks on torsionfreeness and its applications

In this article, we shall characterize torsionfreeness of modules with respect to a semidualizing module in terms of the Serre's condition (S-n). As its applications, we give a characterization of Cohen-Macaulay rings R such that R-? is Gorenstein for all prime ideals ? of height less than n, and we will give a partial answer of Tachikawa conjecture and Auslander-Reiten conjecture.