$(R,m)$ 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)