In this paper, we focus on n-syzygy modules and the injective cogenerator
determined by the minimal injective resolution of a noether ring. We study the
properties of n-syzygy modules and a category Rnβ(modR) which includes
the category consisting of all n-syzygy modules and their applications on
Auslander-type rings. Then, we investigate the injective cogenerators
determined by the minimal injective resolution of R. We show that R is
Gorenstein with finite self-injective dimension at most n if and only if \id
R\leq n and \fd \bigoplus_{i=0}^n I_i(R)< \infty. Some known results can be
our corollaries