The following questions seem basic, but I can't find them in the literature. I'm also unable to think of a counterexample.


Let $A$ be a local Cohen-Macaulay ring of dimension $d$.


1. Let $I$ be an ideal generated by $r$ elements. Is is true that the depth of $A/I$ is at least $d-r$?
2. Let $Q$ be a minimal prime of $A$. Is it true that $A/Q$ is also Cohen-Macaulay?



Two questions about Cohen-Macaulay rings
Nice questions! The answers are no in both cases, although the examples are more interesting than one would expect.


1) Even when $A$ is regular, one can always find an ideal $I$ with $3$ generators such that $A/I$ has depth $0$. This is due to a very nice [result](http://www.ams.org/mathscinet/search/publdoc.html?pg1=IID&s1=42575&vfpref=html&r=81&mx-pid=399074) by Bruns, which says you can construct $3$-generated ideal with all kinds of homological patern. The details are explained in this [answer](https://mathoverflow.net/questions/9352/depth-zero-ideals-in-the-homogenized-weyl-algebra/9357#9357).


2) Let $R=k[X^4,X^3Y,XY^3,Y^4]$. Then $R$ is a domain of dimension $2$ which is not Cohen-Macaulay. So one can write $R=S/Q$, where $R=k[a,b,c,d]$ and $Q$ is a prime ideal of height 2. Take $(f,g)$ to be a regular sequence in $Q$ and let $I=(f,g)$. Then $A=S/I$ is Cohen-Macaulay (being a complete intersection), but $Q$ is a minimal prime of $A$ and $R=S/Q$ is not CM. 


