# Does $fd(M)\lt \infty$ and $id(M)\lt \infty$ imply that $R$ is Gorenstein?

$(R,m)$ is a local Noetherian ring. $M$ is a nonzero finite $R$-module of finite injective dimension($id$). It is known that if $R$ is Gorenstein, then $M$ has finite flat dimension ($fd$). I wonder if the converse is true? So the question is:

Does $fd(M)\lt \infty$ and $id(M)\lt \infty$ imply that $R$ is Gorenstein? ($M$ is a non-zero finite $R$-module)

Yes, even with no finiteness assumption on $M$. See Bourbaki, Algèbre Commutative X, §8, exercise 8 c).