Artinian 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 Artinian is stronger than the property of being Cohen-Macaulay.

Verbal statement

Any Artinian ring is Cohen-Macaulay.

Definitions used

Artinian ring

Further information: Artinian ring

An Artinian ring is a commutative unital ring in which any descending chain of ideals stabilizes after a finite stage.

Cohen-Macaulay ring

Further information: Cohen-Macaulay ring

A Cohen-Macaulay ring is a ring in which, for every maximal ideal, the depth equals the codimension.

Facts used

• In an Artinian ring every element is invertible or a zero divisor: This follows by constructing the descending chain of principal ideals generated by powers of the element.

Proof

In an Artinian ring, every prime ideal is maximal, so in particular Artinian rings are zero-dimensional. Thus, the codimension of any maximal ideal is zero. Hence, we need to prove that the depth of any maximal ideal is zero.

The trick here is to use the Artinianness condition to show that every element of the ring is either invertible or a zero divisor. In particular, any element contained in a maximal ideal must be a zero divisor, and hence, there cannot be any regular sequence of positive length in a maximal ideal.