Note on (weak) Gorenstein global dimensions

In this note we characterize the (resp., weak) Gorenstein global dimension for an arbitrary ring. Also, we extend the well-known Hilbert's syzygy Theorem to the weak Gorenstein global dimension and we study the weak Gorenstein homological dimensions of direct product of rings, which gives examples of non-coherent rings of finite Gorenstein dimensions $>0$ and infinite classical weak dimension.