For a noetherian ring $R$, let $x\in R$ be a non-zerodivisor of an $R$-module $M$. Then this is a well-known fact:
$$\dim M/xM \leq \dim M-1.$$
I saw a proof for the case of finitely generated $M$. I wonder if there is a proof of this fact in the general case.
And, perhaps, is it possible to prove this fact using Krull's Principal Ideal Theorem?