I came across the following question in Hungerford's Algebra:
An integral domain $R$ is a UFD iff every non-zero prime ideal contains a nonzero prime principal ideal.
The forward direction is easy. However, I still don't know how to show the backward direction. I know that I need to show the following statement:
Let $\{a_n\}_{n\in Z^+}$ be a sequence in $R$ such that:$$(a_1)\subseteq(a_2)\subseteq(a_3)\subseteq.....$$Then $\exists N\in \mathbb{Z^+}\forall n\geq N[(a_n)=(a_N)]$
But I couldn't figure out how to prove it. Any hints would be appreciated.
Thank you
