Cohen-Macaulay ring
= Cohen-Macaulay ring
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A commutative Noetherian ring is Cohen-Macaulay when every localization at a prime has <depth of a Noetherian local ring> equal to its Krull dimension. A regular local ring is Cohen-Macaulay, and quotienting one by a nonzero divisor preserves Cohen-Macaulayness while lowering dimension and depth by one. In particular a hypersurface in a regular ambient variety satisfies the depth condition $(S_2)$ in the <Serre criterion for normality>.