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.