Global dimension of quasi Frobenius ring

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.

Solution 1:

Suppose you have a module $M$ with finite projective dimension, say $n$: $$0\to P_n\to \dots\to P_0\to M\to 0.$$ Look at the injection $0\to P_n\to P_{n-1}$. You have that $P_n$ is injective, hence $P_n$ is a direct summand of $P_{n-1}$. But then you could leave out the summand $P_n$ in both $P_n$ and $P_{n-1}$, contradiction to projective dimension $n$.