In the book Cohen-Macaulay rings by Bruns and Herzog. Part of theorem 2.1.2 states
Let $(R,m)$ be a Noetherian local ring, and $M\neq0$ a Cohen-Macaulay $R$-module. Then $\dim R/p=\operatorname{depth}M$ for all $p\in\operatorname{Ass}M$.
If $M=R$ and if $R/p$ is Cohen-Macaulay, then we would have $\operatorname{depth}R/p=\operatorname{depth}R$.
So, does there exist such C-M ring $(R,m)$ with some $p\in \operatorname{Ass}R$ so that $R/p$ is not Cohen-Macaulay? (I am especially curious about the case where $R$ is a Cohen-Macaulay affine ring (i.e. $R=k[x_1,...,x_n]/I$ for some ideal $I\in k[x_1,...,x_n]$ where $k$ is a field) if we don't assume $R$ to be local.)