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.
Read more about Gorenstein Ring: Examples, Properties
Famous quotes containing the word ring:
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