Does the minimal injective resolution have the smallest length?

Question

Let $A$ be a Noetherian (not necessarily local) ring and $M$ a finitely generated $A$-moduel. Is the length of the minimal injective resolution of $M$ always equal to the injective dimension of $M$? (Just like the projective dimension and minimal free resolution.) I suspect the formula for the Bass number $$\mu_i(\mathfrak{p},M)=\mbox{dim}_{\kappa(\mathfrak{p})}\mbox{Ext}^i_{A_{\mathfrak{p}}}(\kappa(\mathfrak{p}),M_{\mathfrak{p}})$$ might hold the key, but I can't seem to go anywhere.

Answer 1

If $$0\rightarrow M\rightarrow Q^0\rightarrow Q^1\rightarrow\cdots\equiv\mathcal{Q}^{\bullet}$$ is any injective resolution $\mathcal{I}^{\bullet}$ of $M$, there is an injective chain map $\mathcal{I}^{\bullet}\hookrightarrow\mathcal{Q}^{\bullet}$, implying that $\mbox{length}(\mathcal{I}^{\bullet})\leq\mbox{length}(\mathcal{Q}^{\bullet})$.