# Algebra switching – Example of a local ring that is not CM and an MCM module on it

I am looking for an example of a local commutative noeterian ring $$(A, m)$$,
and a maximum module of Cohen-Macaulay $$M$$ more than $$A$$ (in particular $$M$$ is finely generated on $$A$$), as for some $$p in Spec (A)$$we have that $$M_p = 0$$,
and $$A_p$$ Nor is it a Cohen-Macaulay ring. Are there any such examples?