Equivalence of definitions of Cohen-Macaulay ring

This article gives a proof/explanation of the equivalence of multiple definitions for the term Cohen-Macaulay ring

View a complete list of pages giving proofs of equivalence of definitions

The definitions that we have to prove as equivalent

The ring is assumed to be Noetherian.

1. For every maximal ideal, the depth equals the codimension
2. For every prime ideal, the depth equals the codimension
3. For every ideal, the depth equals the codimension

Note that (3) implies (2) implies 1, so we need to show that (1) implies (3). In other words, we need to show that assuming depth = codimension for maximal ideals is enough to show that depth = codimension for all ideals.