Let $\Lambda$ and $\Gamma$ be artin algebras and $_{ \Lambda}U_{ \Gamma}$ a faithfully balanced selforthogonal bimodule. In this paper, we first introduce the notion of $k$-Gorenstein modules with respect to $_{ \Lambda}U_{ \Gamma}$ and then characterize it in terms of the $U$-resolution dimension of some special injective modules and the property of the functors ${ \rm Ext}^{ i}({ \rm Ext}^{ i}(-, U), U)$ preserving monomorphisms, which develops a classical result of Auslander. As an application, we study the properties of dual modules relative to Gorenstein bimodules. In addition, we give some properties of $_{ \Lambda}U_{ \Gamma}$ with finite left or right injective dimension.
Categories: math.RA math.RT