Assume $R$ is a commutative local Noetherian ring. It is known that if there is a finitely generated module with finite injective dimension then $R$ is Cohen-Macaulay. My question is:
if $R$ is CM is there a finitely generated module with finite injective dimension?
I think I read the construction of such module in Bruns-Herzog but I cannot remember it nor I can find it on the book. Of course if the ring is a homomorphic image of a Gorenstein ring the you can just take the canonical module, but in general is there such module?