Principal Ideal Domain Modulo a Power of a Prime Ideal

Question

If $R$ is a PID, $p$ is a prime element of $R$, $R/(p)$ is finite, and $\alpha$ is a positive integer, is it true that $\vert R/(p^{\alpha})\vert=\vert R/(p)\vert^{\alpha}$? I seem to recall seeing this somewhere, and it seems to be true intuitively, but I'm not sure.

Answer 1

Yes. This follows from the exact sequence $$0 \to R/p \xrightarrow{p^n} R/p^{n+1} \to R/p^n \to 0.$$