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 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 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
From the given definitions, it is clear that we need to prove that in a zero-dimensional ring, the depth of any prime ideal is zero. This translates to the following:
Given: A commutative unital ring , an ideal <math>I<
