Suppose that $R$ is a commutative ring with unity in which every maximal ideal is finitely generated. Then is $R$ Noetherian?
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http://math.stackexchange.com/questions/183199/commutative-non-noetherian-rings-in-which-all-maximal-ideals-are-finitely-genera and the answer is no, of course. – Timbuc Feb 05 '15 at 11:51
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the answer is no. but by addition assumption, "every prime ideal is finitely generated", it will be yes: see for example here – user 1 Feb 05 '15 at 15:59