Definition 1

A ring $A$ is said to be a Gorenstein ring if it is noetherian and if the $A_m$-module $A_m$ is of finite injective dimension for every maximal ideal $m$ of $A$.

For a noetherian local ring $A$ to be a Gorenstein ring, it is necessary and sufficient that $\mathrm{di}_A(A)$ be finite; for a noetherian ring $A$ to be a Gorenstein ring, it is necessary and sufficient that this be so for $A_m$ for every maximal ideal $m$ of $A$.