A Noetherian local ring \(R\) is Gorenstein if \(R\), as a module over itself, has finite injective dimension. A commutative Noetherian ring \(R\) is Gorenstein if its localization \(R_{\mathfrak p}\) is a Gorenstein local ring for every prime ideal \(\mathfrak p\subseteq R\) [@Eis95].
A commutative Noetherian ring of finite injective dimension over itself is Gorenstein; conversely, a Gorenstein ring of finite Krull dimension has finite injective dimension over itself [@Eis95].