if R is a commutative ring in which all the prime ideals are finitely generated then R is Noetherian [duplicate]

algebraic-geometry abstract-algebra ring-theory commutative-algebra ideals

Prove that if $R$ is a commutative ring in which all the prime ideals are finitely generated, then $R$ is Noetherian.

Here is what I been told to do: Suppose that $R$ is not Noetherian, and use Zorn’s lemma to obtain a maximal element $I$ in the collection of all ideals of $R$ that are not finitely generated. Then use the following proposition: Let $I$ be an ideal of a commutative ring $R$, and let $r ∈ R$. If the ideals $I +rR$ and {$s ∈ R : sr ∈ I$} are finitely generated, then $I$ is a finitely generated ideal.

Can anyone help please. thanks a lot.


Book: Steps in Commutative Algebra (by Sharp)

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