If $R$ is a principal ideal domain and $P \in Spec(R)$, why is $P^mR/P^n=0$ for $m \geq n$?

Question

If $R$ is a principal ideal domain and $P \in Spec(R)$, why is $P^mR/P^n=0$ for $m \geq n$? This is a step on a proof I’m trying to understand.

Answering to the question in the comments:

The context is: we want to show that $R/P^m \otimes_R R/P^n $ is isomorphic to $R/P^n$ as $R$-modules, so using that $P^mR/P^n=0$ we get

$$R/P^m \otimes_R R/P^n \cong R/P^n/P^mR/P^n \cong R/P^n$$

(because we know that if $M$ is a $R$-module, and $I \subset R$ an ideal, we have a canonical isomorphism of $R$-modules $(R/I) \otimes_R M \cong M/IM$, and here we take $I:=P^m$ and $M:=R/P^n$).

Answer 1

Simply note that $P^m\subseteq P^n$ for $n\leq m$ - this is true for any ideal in any ring. Then $P^mR\subseteq P^n$, so $P^mR/P^n = 0$ by the definition of the quotient.