I have following task:
Given a factorial ring $R$ (i.e. with unique prime decomposition) and a prime element $a \in R$ prove that if $I \subseteq R$ is a prime ideal with $(0) \subseteq I \subseteq (a)$, then $I=(0)$ or $I=(a)$.
So far I assumed that $I \neq (0)$ and $I \neq (a)$ in order to find a contradiction.
I was able to prove that no power $a^s$ of $a$ (for $s \in \mathbb N$) can be contained in $I$, so $a^s \not\in I\ \forall s \in\mathbb N$, but I'm stuck there.
Is this the right way to go? Or is there another easier way that I missed?
I'm glad for any hints or alternative solutions.