Gorenstein ring

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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.

A noteworthy occurrence of the concept is as one ingredient (among many) of the solution by Andrew Wiles to the Fermat Conjecture.


  1. Every complete intersection ring is Gorenstein.
  2. Every regular local ring is a complete intersection ring, so is Gorenstein.

See also Daniel Gorenstein.