In commutative algebra, a perfect ideal is a proper ideal in a Noetherian ring such that its grade equals the projective dimension of the associated quotient ring.[1]
A perfect ideal is unmixed.
For a regular local ring a prime ideal is perfect if and only if is Cohen-Macaulay.
The notion of perfect ideal was introduced in 1913 by Francis Sowerby Macaulay[2] 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[3] point out, Macaulay's original definition of perfect ideal coincides with the modern definition when is a homogeneous ideal in polynomial ring, but may differ otherwise. Macaulay used Hilbert functions to define his version of perfect ideals.
References
🔥 Top keywords: Main PageSpecial:SearchPage 3Wikipedia:Featured picturesHouse of the DragonUEFA Euro 2024Bryson DeChambeauJuneteenthInside Out 2Eid al-AdhaCleopatraDeaths in 2024Merrily We Roll Along (musical)Jonathan GroffJude Bellingham.xxx77th Tony AwardsBridgertonGary PlauchéKylian MbappéDaniel RadcliffeUEFA European Championship2024 ICC Men's T20 World CupUnit 731The Boys (TV series)Rory McIlroyN'Golo KantéUEFA Euro 2020YouTubeRomelu LukakuOpinion polling for the 2024 United Kingdom general electionThe Boys season 4Romania national football teamNicola CoughlanStereophonic (play)Gene WilderErin DarkeAntoine GriezmannProject 2025