$(R,m)$ 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: 



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> Does $fd(M)\lt \infty$ and $id(M)\lt \infty $ imply that $R$ is Gorenstein? ($M$ is a non-zero finite $R$-module) 
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Does $fd(M)\lt \infty$ and $id(M)\lt \infty $ imply that $R$ is Gorenstein?
Yes. As you have stated it, this is a theorem of H.-B. Foxby (Math. Scand. 40 (1977), 5-19, "Isomorphisms between complexes with applications to the homological theory of modules." <http://www.mscand.dk/article/download/11671/9687>) Actually, Foxby says "projective dimension" instead of "flat dimension", but for finite modules over Noetherian local rings these invariants coincide.


