= Cohen-Macaulay ring
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A commutative <Noetherian ring> is Cohen-Macaulay when every <localization> at a <prime ideal> has <depth of a Noetherian local ring> equal to its <Krull dimension>. A <regular local ring> is Cohen-Macaulay. If $(R,\mathfrak m)$ is a Cohen-Macaulay <local ring> and $x\in\mathfrak m$ is a <non-zero-divisor>, then $R/(x)$ is Cohen-Macaulay and both its <depth of a Noetherian local ring> and <Krull dimension> are one less than those of $R$.
The requirement $x\in\mathfrak m$ excludes <units>, whose quotient would be the zero <ring>. In particular, a <hypersurface> in a regular ambient <variety> satisfies the depth condition $(S_2)$ in the <Serre criterion for normality>.
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