My professor asks me to give an example about a local non-CM rings that are CM after modding out at any minimal prime. After a while failing to do so, I believe it is impossible since there exists a minimal prime $P$ s.t. $\dim R/P=\dim R$. So the dimension can be unchanged, hence there is no way $\dim R/P =depth(R/P)$. Am I wrong somewhere? If I am, can you show me an example?
Speaking of this topic, I also want to know is there a ring that is CM but not after modding out at minimal primes.
Is there an intuition behind all this? I learned about Krull diension and depth with pure algebraic definitions, so I can prove theorems, but cannot visualize such properties. Can you recommend me some books about this?
For the first part of your question, take $R=k[\![x,y]\!]/(x^2,xy)$. Then $R$ is not Cohen-Macaulay (it has dimension $1$ and depth $0$) and has only one minimal prime, namely, $(x)$. But $R/xR \cong k[\![y]\!]$ which is Cohen-Macaulay. Of course, $R/xR$ has the same dimension as $R$, but has larger depth than $R$.
For the second part of your question, killing minimal primes in a Cohen-Macaulay local ring frequently results in non-Cohen Macaulay rings, even in nice situations. For a concrete example, I'll point to Takumi Murayama's answer to a similar question.
Cohen-Macaulayness arises naturally from many different but related angles. In a loose sense, the dimension is a geometric invariant while the depth is an algebraic one. When these coincide, the geometry is a good (or at least better) reflection of what's happening algebraically. For a good overview, I would recommended starting from near the end of page 896 in this research paper1 of Mel Hochster. He makes a point of giving some insight into the Cohen-Macaulay condition; in particular several examples are provided and discussed.
1Melvin Hochster. "Some applications of the Frobenius in characteristic 0." Bull. Amer. Math. Soc. 84 (5) 886 - 912, September 1978.