Syzygy Modules and Injective Cogenerators for Noether Rings

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 $R_n(\mod R)$ 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.