Let $R$ be a quasi-Frobenius ring (so $R$ is self-injective and left and right noetherian). I want to prove that $lD(R)=0$ or $\infty$, where $lD(R)$ denotes the left global dimension.
I'm unsure about how to go about proving this; the only thing I can think of is to somehow show that if we assume that some $R$ module $A$ has a finite projective resolution then it is in fact projective and hence has global dimension $0$. I tried to do this using the fact that a module over a quasi Frobenius ring is projective if and only if it is injective, but I didn't get far.