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.

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    $\begingroup$ It seems impossible that this question has not been asked before because the statement is so reasonable. Oh, look! : ) $\endgroup$ – Nobody Aug 5 '16 at 16:48
  • $\begingroup$ @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 $\endgroup$ – user228169 Aug 6 '16 at 8:00
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    $\begingroup$ "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. $\endgroup$ – Nobody Aug 6 '16 at 14:10
  • $\begingroup$ Related: math.stackexchange.com/questions/1818783/… $\endgroup$ – user26857 Aug 26 '16 at 10:48
  • $\begingroup$ So... in a Noetherian ring, why do you need Zorn's Lemma to conclude that some collection of ideals has a maximal element? $\endgroup$ – mathguy Aug 27 '16 at 12:31

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