{{Short description|Type of ideal relevant for Noetherian rings}}{{Context|date=August 2023}}

In commutative algebra, a '''perfect ideal''' is a proper ideal <math>I</math> in a Noetherian ring <math>R</math> such that its grade equals the projective dimension of the associated quotient ring.<ref name="Matsumura1">{{cite book |last=Matsumura |first=Hideyuki |author-link=Hideyuki Matsumura |date=1987 |title=Commutative Ring Theory |url=https://www.cambridge.org/core/books/commutative-ring-theory/02819830750568B06C16E6199F3562C1 |location=Cambridge |publisher=Cambridge University Press | page=132 |isbn=9781139171762}}</ref>

<math>\textrm{grade}(I)=\textrm{proj}\dim(R/I).</math>

A perfect ideal is unmixed.

For a regular local ring <math>R</math> a prime ideal <math>I</math> is perfect if and only if <math>R/I</math> is Cohen-Macaulay.

The notion of perfect ideal was introduced in 1913 by Francis Sowerby Macaulay<ref name=Macauley>{{cite journal |last=Macaulay |first=F. S. |author-link=Francis Sowerby Macaulay |date=1913 |title=On the resolution of a given modular system into primary systems including some properties of Hilbert numbers |url=https://link.springer.com/article/10.1007/BF01455345 |journal=Math. Ann. |volume=74 |issue=1 |pages=66–121 |doi=10.1007/BF01455345 |s2cid=123229901 |access-date=2023-08-06|url-access=subscription }}</ref> in connection to what nowadays is called a Cohen-Macaulay ring, but for which Macaulay did not have a name for yet. As Eisenbud and Gray<ref name=EisenbudGray>{{cite journal |last1=Eisenbud |first1=David |author-link1=David Eisenbud |last2=Gray |first2=Jeremy |author-link2=Jeremy Gray (mathematician) |date=2023 |title=F. S. Macaulay: From plane curves to Gorenstein rings |url=https://www.ams.org/journals/bull/2023-60-03/S0273-0979-2023-01787-4/ |journal=Bull. Amer. Math. Soc. |volume=60 |issue=3 |pages=371–406 |doi=10.1090/bull/1787 |access-date=2023-08-06|doi-access=free }}</ref> point out, Macaulay's original definition of perfect ideal <math>I</math> coincides with the modern definition when <math>I</math> is a homogeneous ideal in a polynomial ring, but may differ otherwise. Macaulay used Hilbert functions to define his version of perfect ideals.

==References== {{Reflist}}

Category:Ideals (ring theory) Category:Commutative algebra