Distinguishing dualizing modules among semidualizing modules

Let C be a semidualizing R-module. We present some necessary and sufficient conditions for C to be dualizing in several directions. First, we show that vanishing of appropriate Bass numbers(resp. dual Bass numbers) on the class FC (resp. LC) is equivalent with the dualizing property of C. Next, we show that C is dualizing if and only if the class LC is closed with respect to torsion product. Finally, we define two new notions, namely n-C-perfect and n-C-coperfect modules, and give a characterization for dualizing modules in the case where (R,m) is a ddimensional local ring, as follows: C is dualizing if and only if E(R/m) (resp. bR) is d-C-perfect (resp. d-C-coperfect)