Let $R$ be a commutative Noetherian local ring (having unity) with the maximal ideal $m$, which consists only of zero-divisors. Then for any finitely generated $R$-module $M$, the projective dimension $\operatorname{pd}(M)$ of $M$ is zero or infinity.
I am glitched at this problem having searched of help. I would appreciate the solver. Of course, I am aware that as of a left Noetherian ring $S$ for which the $pd(M)$ of any f.g. right module $M$ is zero or infinity, $S$ is a left Kasch ring. Thanks in advance!
Hint. If there's a module $M$ with $0<\operatorname{pd}(M)<\infty$, then consider a resolution $$0\to F_m\to F_{m-1}\to\dots\to F_0\to M\to 0$$ of $M$ by finitely generated free modules. Try to prove that the injective map $F_m\to F_{m-1}$ must split and that therefore there's a shorter resolution.