  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_p is Gorenstein for all prime ideals p of height less than n, and we
will give a partial answer of Tachikawa conjecture and Auslander-Reiten
conjecture.
