Direct product of Cohen-Macaulay rings/Eisenbud, Exercise 18.6 
Somehow I believe (or doubt (!)) that direct product of two Cohen-Macaulay (C-M) rings may not be C-M. Can anybody give me an example verifying this? 

I would be grateful to him/her.

Any property of rings of the form "every local ring is blah" (such as Cohen-Macaulay) is going to pass to finite direct products of rings with the property. This is because any local ring of the direct product of two rings may be identified with a local ring of one of the factors (geometrically, $\mathrm{Spec}(A\times B)=\mathrm{Spec}(A)\coprod\mathrm{Spec}(B)$, and this is the usual disjoint union of topological spaces).
