Can you offer some examples of such rings, other than $\frac{k[x,y]}{(x^{2}, xy)}$. Thanks.



Examples of one-dimensional non-Cohen Macaulay rings
In dimension $1$, Cohen-Macaulay just mean *unmixed*, so all the associated primes have the same dimension. Thus the easiest way to cook up a non-CM ring of dimension $1$ is: Pick your favorite regular ring (say $A=k[x,y,z]$). Take an ideal of dimension $1$, say $I=(x,y)$. Take another ideal of dimension $0$, say $J=(x^3,y^4,z^5)$ such that $J$ is not contained in $I$. Now take $R=A/(I\cap J)$. Geometrically we just throw 2 things of pure dimensions $1$ and $0$ together. In some sense, all non-CM rings of dimension $1$ arise this way. 


