Gorenstein Ring

Gorenstein Ring

In commutative algebra, a Gorenstein local ring is a Noetherian commutative local ring R with finite injective dimension, as an R-module. There are many equivalent conditions, some of them listed below, most dealing with some sort of duality condition.

A Gorenstein commutative ring is a commutative ring such that each localization at a prime ideal is a Gorenstein local ring. The Gorenstein ring concept is a special case of the more general Cohen–Macaulay ring.

The classical definition reads:

A local Cohen–Macaulay ring R is called Gorenstein if there is a maximal R-regular sequence in the maximal ideal generating an irreducible ideal.

For a Noetherian commutative local ring of Krull dimension, the following are equivalent:

• has finite injective dimension as an -module;
• has injective dimension as an -module;
• for and is isomorphic to ;
• for some ;
• for all and is isomorphic to ;
• is an -dimensional Gorenstein ring.

A (not necessarily commutative) ring R is called Gorenstein if R has finite injective dimension both as a left R-module and as a right R-module. If R is a local ring, we say R is a local Gorenstein ring.