One-dimensional Noetherian domain 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

Property-theoretic statement

The property of commutative unital rings of being a one-dimensional Noetherian domain is stronger than the property of being a Cohen-Macaulay ring.

Verbal statement

Any one-dimensional Noetherian domain (i.e. a Noetherian domain whose Krull dimension is exactly one) is Cohen-Macaulay.

Proof

By definition, every maximal ideal has codimension one.

Because any nonzero element is a nonzerodivisor, we can clearly, for every maximal ideal, pick a regular sequence of length one. Thus, the depth of any maximal ideal is one.

Hence, for any maximal ideal, the depth equals the codimension, so the ring is Cohen-Macaulay.