Why $R/J$ has finite ideals if $R$ PID?

Question

I have this problem:

Let $R$ a PID and $J\subseteq R$ a non-zero ideal. Prove that $R/J$ has a finite number of ideals.

I think that I must use the biyective correspondance between the ideals of $R$ that contain $J$ and the ideals of $R/J$, but I still don't see why I have a finite number of ideals that contain $J$. Can anyone give me a hint or I'm going in the wrong way?

Answer 1

Since $R$ is a PID, any ideal in $R$ is principal. Let $J=(x)$ and let $I=(y)$ be another ideal. Then $J\subseteq I$ if and only if $y$ divides $x$.

Can you show that there are (up to multiplication by units) only finitely many elements of $R$ that divide $x$?