Wakamatsu Tilting Modules, $U$-Dominant Dimension and $k$-Gorenstein Modules

Abstract

Let $Λ$ and $Γ$ be left and right noetherian rings and $_ΛU$ a Wakamatsu tilting module with $Γ={\rm End}(_ΛT)$. We introduce a new definition of $U$-dominant dimensions and show that the $U$-dominant dimensions of $_ΛU$ and $U_Γ$ 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.