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 $\mathcal{F}_C$ (resp. $ \mathcal{I}_C $) is equivalent with the dualizing property of $C$. Next, we show that $C$ is dualizing if and only if the class $ \mathcal{I}_C $ 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 $(R,\mathfrak{m})$ is a $d$-dimensional local ring, as follows: $C$ is dualizing if and only if $E(R/\mathfrak{m})$ (resp. $\widehat{R}$) is $d$-$C$-perfect (resp. $d$-$C$-coperfect)