Perfect ideal

In commutative algebra, a perfect ideal is a proper ideal ${\displaystyle I}$ in a Noetherian ring ${\displaystyle R}$ such that its grade equals the projective dimension of the associated quotient ring.¹

${\displaystyle {\textrm {grade}}(I)={\textrm {proj}}\dim(R/I).}$

A perfect ideal is unmixed.

For a regular local ring ${\displaystyle R}$ a prime ideal ${\displaystyle I}$ is perfect if and only if ${\displaystyle R/I}$ is Cohen-Macaulay.

The notion of perfect ideal was introduced in 1913 by Francis Sowerby Macaulay² 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 Gray³ point out, Macaulay's original definition of perfect ideal ${\displaystyle I}$ coincides with the modern definition when ${\displaystyle I}$ is a homogeneous ideal in a polynomial ring, but may differ otherwise. Macaulay used Hilbert functions to define his version of perfect ideals.