On X -Gorenstein projective dimensions and precovers

Abstract

For a class of R-modules X containing all projective R-modules, the X -Gorenstein projective R-modules vary from projective to Gorenstein projective R-modules. We characterize the rings over which the left global X - Gorenstein projective dimensions are finite. If further Y contains all injective R-modules, we show the existence of a new left global Gorenstein dimension of R with respect to X and Y satisfying proper conditions. As an application we characterize Ding-Chen rings by this new global Gorenstein dimension and show the existence of Ding-Chen rings with infinite global Gorenstein dimension. We also show the existence of X -Gorenstein projective precovers for a large class of rings.