Let $(R,\mathfrak{m})$ be a Noetherian local ring. We recall the depth of $R$ is equal to the length of any maximal $\mathfrak{m}$-regular sequence.
My question is: Is it true that $$\dim R = \max_{\mathfrak{p} \in \operatorname{Ass}(R)} \dim R/\mathfrak{p}$$ and $$\operatorname{depth} R = \min_{\mathfrak{p} \in \operatorname{Ass}(R)} \dim R/\mathfrak{p}\ ?$$
If true, this would give a very nice geometric meaning to Cohen-Macaulay rings: The closed subschemes corresponding to associated primes are all equidimensional.
The dimension formula it is more or less a definition since minimal primes are associated.
For the depth formula, take a domain which is not Cohen-Macaulay. This gives a contradiction.