  Let $\Lambda$ and $\Gamma$ be left and right noetherian rings and
$_{\Lambda}U$ a Wakamatsu tilting module with $\Gamma ={\rm End}(_{\Lambda}T)$.
We introduce a new definition of $U$-dominant dimensions and show that the
$U$-dominant dimensions of $_{\Lambda}U$ and $U_{\Gamma}$ are identical. We
characterize $k$-Gorenstein modules in terms of homological dimensions and the
property of double homological functors preserving monomorphisms. We also study
a generalization of $k$-Gorenstein modules, and characterize it in terms of
some similar properties of $k$-Gorenstein modules.
