In an integral domain $R$ the prime ideals are principal ideals. (given)
So my attempt was using the Zorn lemma. We assume that there exists a chain of non principal ideals so by set inclusion the chain will be partially ordered by set inclusion.
Now the chain will have a maximal element. My reason is like this that let $I_1$ be the ideal generated by $(a, b) $ , then this ideal will be contained in the ideal $I_2=(a, b, c) $ so I think that it will have a upper bound which will be the whole ring itself. Now by Zorn lemma there will be a maximal element. Is this OK?