Cohen-Macaulay ring

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.