Here is Bruns and Herzog's book Cohen-Macaulay Rings, Theorem 3.1.17:
Let $R$ be a Noetherian local ring, and $M$ a finite $R$-module of finite injective dimension. Then $\operatorname{inj\ dim}M = \operatorname{depth}R$.
Here are my questions:
1) It seems that $\operatorname{inj\ dim}M$ is independent of $M$, how to understand this? Here I do not want an exact proof, but want to make it more nature, maybe some good examples can do this.
2) If $M$ has infinite injective dimension, is $\operatorname{depth}R$ also infinite?
3) If $\operatorname{depth}R$ is finite, $M$ has finite or infinite injective dimension, do the result also true?
1) The injective dimension (i.d) is not independent of $M$. According to what $M$ is, we have two possibilities: either i.d. is finite or it is infinite. In the first case it is equal to the depth.
2) The dimension of local Noetherian rings is finite. Also, the depth is bounded above by the dimension (see Theorem 1.2.12 in B&H). Hence a local Noetherian ring has always finite depth.
3) The depth is always finite.