Homological behavior of Auslander's $k$-Gorenstein rings

In this paper we mainly study the homological properties of dual modules over $k$-Gorenstein rings. For a right quasi $k$-Gorenstein ring $\Lambda$, we show that the right self-injective dimension of $\Lambda$ is at most $k$ if and only if each $M \in$mod $\Lambda$ satisfying the condition that Ext$_{\Lambda}^i(M, \Lambda)=0$ for any $1\leq i \leq k$ is reflexive. For an $\infty$-Gorenstein ring, we show that the big and small finitistic dimensions and the self-injective dimension of $\Lambda$ are identical. In addition, we show that if $\Lambda$ is a left quasi $\infty$-Gorenstein ring and $M\in$mod $\Lambda$ with grade$M$ finite, then Ext$_{\Lambda}^i($Ext$_{\Lambda ^{op}}^i($Ext$_{\Lambda}^{{\rm grade}M}(M, \Lambda), \Lambda), \Lambda)=0$ if and only if $i\neq$grade$M$. For a 2-Gorenstein ring $\Lambda$, we show that a non-zero proper left ideal $I$ of $\Lambda$ is reflexive if and only if $\Lambda /I$ has no non-zero pseudo-null submodule.