One-dimensional Noetherian domain implies Cohen-Macaulay

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.