Zero-dimensional Noetherian implies Cohen-Macaulay

This article gives the statement and possibly, proof, of an implication relation between two commutative unital ring properties. That is, it states that every commutative unital ring satisfying the first commutative unital ring property must also satisfy the second commutative unital ring property
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Statement

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.

Definitions used

Zero-dimensional ring

Further information: 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

Further information: 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

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.