Injective Maximal Cohen-Macaulay modules

Question

Let $R$ be a Gorenstein (not necessarily commutative) ring and let $I$ be an injective finitely generated module over $R$. Is it true that if $\operatorname{Ext}_R^i(I, R)=0$ for $i > 0$, then $I$ is projective?

Answer 1

If $R$ is commutative, then the answer is yes and we don't need the vanishing of those Exts.

Actually we will prove the following

Proposition. Let $R$ be a commutative Gorenstein ring and $I$ a finitely generated injective $R$-module. Then $I$ is projective.

We can consider $R$ local. The existence of a nonzero finitely generated injective $R$-module implies that $R$ is artinian; see Bruns and Herzog, 3.1.23. On the other side over Gorenstein local rings finitely generated modules of finite injective dimension have also finite finite projective dimension; Bruns and Herzog, 3.1.25 or look here. Now simply apply the Auslander-Buchsbaum formula and deduce that the projective dimension of $I$ is $0$, i.e. $I$ is free.