Perfect ideal

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

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

A perfect ideal is unmixed.

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

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