2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/100859Let $Λ$ 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.24 pages. Lect. Notes in Pure and Applied Math. 249, 2006. Some misprints in Theorems 5.2 and 5.4 are corrected (1. In the statement 5.2(2) and the statement 5.4(5): "torsionless" is replaced by "$U$-torsionless". 2. In p.18, line -12: "with $P$ projective" is replaced by "with $P$ in add$_ΛU$".)Rings and AlgebrasRepresentation Theory16E10; 16E30; 16D90text