Let $R$ be a commutative Noetherian ring with unity such that every maximal ideal is principal. Then is it true that every ideal of $R$ is principal ?
So I think I should go like this: Suppose not. Then there is an ideal which is not principal. Consider the collection of all such ideals. It has a maximal element by Zorn's Lemma. If I can show that this maximal element is a maximal ideal then I will have the required contradiction, but I am unable to show that. Is this way correct ? Is there any other way ? Please help. Thanks in advance.