Cohen-Macaulay implies universally catenary

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 Cohen-Macaulay ring is stronger than the property of being a universally catenary ring.

Verbal statement

Any Cohen-Macaulay ring is universally catenary.

Definitions used

Cohen-Macaulay ring

Further information: Cohen-Macaulay ring

Catenary ring

Further information: Catenary ring

Universally catenary ring

Further information: Universally catenary ring

Facts used

• Cohen-Macaulay is polynomial-closed
• Cohen-Macaulay is strongly local

Proof

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References

• Book:Eisenbud, Page 457, Corollary 18.10