  We construct new complete cotorsion pairs in the categories of modules and
chain complexes over a Gorenstein ring $R$, from the notions of Gorenstein
homological dimensions, in order to obtain new Abelian model structures on both
categories. If $r$ is a positive integer, we show that the class of modules
with Gorenstein-projective (or Gorenstein-flat) dimension $\leq r$ forms the
left half of a complete cotorsion pair. Analogous results also hold for chain
complexes over $R$. In any Gorenstein category, we prove that the class of
objects with Gorenstein-injective dimension $\leq r$ is the right half of a
complete cotorsion pair. The method we use in each case consists in
constructing a cogenerating set for each pair. Later on, we give some
applications of these results. First, as an extension of some results by M.
Hovey and J. Gillespie, we establish a bijective correspondence between the
class of differential graded $r$-projective complexes and the class of modules
over $R[x] / (x^2)$ with Gorenstein-projective dimension $\leq r$, provided $R$
is left and right Noetherian with finite global dimension. The same
correspondence is also valid for the (Gorenstein-)injective and
(Gorenstein-)flat dimensions.
