  In this paper, we mainly investigate the $\mathfrak{X}$-Gorenstein projective
dimension of modules and the (left) $\mathfrak{X}$-Gorenstein global dimension
of rings. Some properties of $\mathfrak{X}$-Gorenstein projective dimensions
are obtained. Furthermore, we prove that the (left) $\mathfrak{X}$-Gorenstein
global dimension of ring $R$ is equal to the supremum of the set of
$\mathfrak{X}$-Gorenstein projective dimensions of all cyclic (left)
$R$-modules. This result extends the well-known Auslander's theorem on the
global dimension and its Gorenstein homological version.
