such that its grade equals the projective dimension of the associated quotient ring.
A perfect ideal is unmixed.
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
is a homogeneous ideal in a polynomial ring, but may differ otherwise.
Macaulay used Hilbert functions to define his version of perfect ideals.