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 (Hartshorne 1967)). The name comes from a duality property of singular plane curves studied by Gorenstein (1952) (who was fond of claiming that he did not understand the definition of a Gorenstein ring[citation needed]). The zero-dimensional case had been studied by Macaulay (1934). Serre (1961) and Bass (1963) 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.

Universally catenary ringsCohen–Macaulay ringsGorenstein ringscomplete intersection ringsregular local rings


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 HomR(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.[1] More generally, a Noetherian local ring R is Gorenstein if and only if there is a regular sequence a1,...,an in the maximal ideal of R such that the quotient ring R/( a1,...,an) 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 = kR1 ⊕ ... ⊕ Rm, then R is Gorenstein if and only if it satisfies Poincaré duality, meaning that the top graded piece Rm has dimension 1 and the product Ra × RmaRm is a perfect pairing for every a.[2]

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: RF such that the symmetric bilinear form (x, y) := e(xy) on R (as an F-vector space) is nondegenerate.[3]

For a commutative Noetherian local ring (R, m, k) of Krull dimension n, the following are equivalent:[4]

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.



In the context of graded rings R, the canonical module of a Gorenstein ring R is isomorphic to R with some degree shift.[6]
Namely, a graded domain R is Gorenstein if and only if it is Cohen–Macaulay and the Hilbert series is symmetric in the sense that
for some integer s, where n is the dimension of R.[8]


  1. ^ Eisenbud (1995), Proposition 21.5.
  2. ^ Huneke (1999), Theorem 9.1.
  3. ^ Lam (1999), Theorems 3.15 and 16.23.
  4. ^ Matsumura (1989), Theorem 18.1.
  5. ^ Matsumura (1989), Theorem 18.3.
  6. ^ Eisenbud (1995), section 21.11.
  7. ^ Bruns & Herzog (1993), Theorem 3.5.8.
  8. ^ Stanley (1978), Theorem 4.4.
  9. ^ Eisenbud (1995), Corollary 21.20.
  10. ^ Bruns & Herzog (1993), Theorem 3.4.1.