Zero-dimensional Noetherian implies Cohen-Macaulay

Verbal statement
Any Noetherian zero-dimensional ring (i.e., a ring in which every prime ideal is maximal) is Cohen-Macaulay: for any prime ideal, the depth equals the codimension.

Property-theoretic statement
The property of commutative unital rings of being a Noetherian zero-dimensional ring is stronger than the property of being a Cohen-Macaulay ring.

Zero-dimensional ring
A commutative unital ring is termed zero-dimensional if it satisfies the following equivalent conditions:


 * Every prime ideal is maximal
 * Every prime ideal is a minimal prime
 * Every prime ideal has codimension zero

Cohen-Macaulay ring
A commutative unital ring is termed Cohen-Macaulay if it satisfies the following equivalent conditions:


 * For every prime ideal, the depth equals the codimension. Here, the depth of an ideal is the maximum possible length of a regular sequence in that ideal.
 * For every maximal ideal, the depth equals the codimension.

Proof outline

 * By the definition of zero-dimensional ring, every prime has codimension zero.
 * Thus, by the definition of Cohen-Macaulay ring, it suffices to show that every prime has depth zero i.e. that any element in a minimal prime is a zero divisor.
 * This follows from the fact that in a Noetherian ring, every element in a minimal prime is a zero divisor.