Gorenstein ring

In commutative algebra, a Gorenstein local ring is a commutative Noetherian local ring "R" with finite injective dimension as an "R"-module. There are many equivalent conditions, some of them listed below, often saying that a Gorenstein ring is self-dual in some sense.
Gorenstein rings were introduced by Grothendieck in his 1961 seminar (published in ). The name comes from a duality property of singular plane curves studied by (who was fond of claiming that he did not understand the definition of a Gorenstein ring). The zero-dimensional case had been studied by . and publicized the concept of Gorenstein rings.
Frobenius rings are noncommutative analogs of zero-dimensional Gorenstein rings. Gorenstein schemes are the geometric version of Gorenstein rings.
For Noetherian local rings, there is the following chain of inclusions.
Definitions.
A Gorenstein ring is a commutative Noetherian ring such that each localization at a prime ideal is a Gorenstein local ring, as defined above. A Gorenstein ring is in particular Cohen–Macaulay.
One elementary characterization is: a Noetherian local ring "R" of dimension zero (equivalently, with "R" of finite length as an "R"-module) is Gorenstein if and only if Hom"R"("k", "R") has dimension 1 as a "k"-vector space, where "k" is the residue field of "R". Equivalently, "R" has simple socle as an "R"-module. More generally, a Noetherian local ring "R" is Gorenstein if and only if there is a regular sequence "a"1...,"a""n" in the maximal ideal of "R" such that the quotient ring "R"/( "a"1...,"a""n") is Gorenstein of dimension zero.
For example, if "R" is a commutative graded algebra over a field "k" such that "R" has finite dimension as a "k"-vector space, "R" = "k" ⊕ "R"1 ⊕ ... ⊕ "R""m", then "R" is Gorenstein if and only if it satisfies Poincaré duality, meaning that the top graded piece "R""m" has dimension 1 and the product "R""a" × "R""m"−"a" → "R""m" is a perfect pairing for every "a".
Another interpretation of the Gorenstein property as a type of duality, for not necessarily graded rings, is: for a field "F", a commutative "F"-algebra "R" of finite dimension as an "F"-vector space (hence of dimension zero as a ring) is Gorenstein if and only if there is an "F"-linear map "e": "R" → "F" such that the symmetric bilinear form ("x", "y") := "e"("xy") on "R" (as an "F"-vector space) is nondegenerate.
For a commutative Noetherian local ring ("R", "m", "k") of Krull dimension "n", the following are equivalent:
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, "R" is said to be a local Gorenstein ring.