Gorensteinring

A Gorenstein ring is a ring that is in commutative algebra, a branch of mathematics studied. A Gorenstein ring is a Cohen- Macaulay ring with certain additional properties. A Gorenstein singularity is a singularity whose local ring is a Gorenstein ring.

The rings were named after Daniel Gorenstein, although this always claimed not even to understand the definition.

This article deals with commutative algebra. In particular, all rings considered are commutative and have an identity element. Ring homomorphisms form elements from one to one elements. For more details see Commutative Algebra.

Definition

Is a Noetherian local ring with maximal ideal -dimensional, it is called a lot of a system of parameters of when this quantity - generated primary ideal. (One can show that a Noetherian local ring always has a system of parameters. )

Is

Thus, the number

Regardless of the parameter system.

This number is called the type.

A local Gorenstein ring is a Cohen- Macaulay ring of type 1

A Noetherian ring R is called Gorenstein ring if all its localizations of maximal ideals are local Gorenstein rings.

( This definition follows Kunz 1980. Often a Gorenstein ring of the injective dimension is defined, see below. )

Properties

• Is a local Cohen- Macaulay ring if and so is a Gorenstein ring if the ideal generated by a system of parameters is irreducible. ( An irreducible ideal is an ideal that can not be represented as trivaler average of ideals. )
• A local Noetherian ring if and only a Gorenstein ring if its injective dimension is finite.
• Each local ring complete intersection, a Gorenstein ring. In particular, each regular local ring is a Gorenstein ring.

Examples

• A body, then the variety, which consists of the X-axis and the Y axis is described by the coordinates of the ring.

• The ring is a local -dimensional ring. It is therefore Cohen- Macaulay. He is not Gorenstein, since the zero ideal is indeed -primary, but not irreducible.