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The ring $R = k[x,y]/(x^2, xy)$ is a simple example of a local commutative noetherian ring that is not Cohen-Macaulay. It is sometimes referred to as the "Emmy Ring."

This ring is very useful for showing how unintuitive non-CM rings can be. For instance, letting $I = (x)$, then $\operatorname{depth} R/I = 1 > 0 = \operatorname{depth} R$; in particular the (innocuous looking) inequality

$\operatorname{depth} R/I + \operatorname{grade} I \leq \operatorname{depth} R$

need not hold. Here $\operatorname{grade} I$ is the length of the longest regular sequence in $I$.