example of injective module

Question

Can you provide some examples of a commutative Noetherian local ring $(R,m,k)$ and an $R$-module $M$ that is not finitely generated, such that

$id_R(M)\neq \sup \{i\in \mathbb{N}_0 \mid Ext_R^i(R/m,M)\neq 0\}$?

Here, $id_R(M)$ denotes the injective dimension of the $R$-module $M$.Thanks.

Answer 1

Let $R$ be a regular local ring of dimension two and let $M=R_f$ for some non-zero element $f$ in the maximal ideal. Then, $\mathrm{Ext}^i(k,M)=0$ for all $i$, but $M$ is not injective.