Cohen-Macaulay ring

Symbol-free definition
A commutative unital ring is termed Cohen-Macaulay if it is Noetherian and satisfies the following equivalent conditions:


 * For any maximal ideal, the depth equals the codimension
 * For any prime ideal, the depth equals the codimension
 * For any ideal, the depth equals the codimension

Spectrum
The spectrum of a Cohen-Macaulay ring has the following important geometric property: If two irreducible components intersect, they must have the same dimension. Here, by the dimension of an irreducible component, we mean the dimension of the corresponding minimal prime ideal, or more explicitly, the Krull dimension of the quotient ring by that minimal prime ideal.

This rules out, for instance, rings like $$k[x,y,z]/(xz,yz)$$.

However:


 * The spectrum need not be irreducible: The ring $$k[x,y]/(xy)$$ is Cohen-Macaulay, although its spectrum has two irreducible components.
 * All components of the spectrum need not have the same dimension: For instance, we could have a ring that is a disjoint union of irreducible subsets of different dimensions
 * Not every irreducible space is the spectrum of a Cohen-Macaulay ring. In other words, not every integral domain, or even every affine domain, is Cohen-Macaulay,