# Let $R$ be a commutative Noetherian ring with unity such that every maximal ideal is principal. Then is $R$ a PIR?

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.

• It seems impossible that this question has not been asked before because the statement is so reasonable. Oh, look! : ) – Nobody Aug 5 '16 at 16:48
• @Nobody: first of all the question there is not what I ask , it was only mentioned in the accepted answer . Secondly there is no proof ( at least not the way I am approaching ) in that answer there – user228169 Aug 6 '16 at 8:00
• "Is there any other way?" Yes, theorem 12.3. I'm commenting for a reason; this isn't a full answer addressing every facet of your question. – Nobody Aug 6 '16 at 14:10
• – user26857 Aug 26 '16 at 10:48
• So... in a Noetherian ring, why do you need Zorn's Lemma to conclude that some collection of ideals has a maximal element? – mathguy Aug 27 '16 at 12:31