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Let $\mathbb Z[X]$ be the ring of polynomials in one variable. It is a well-known fact that it is a Noetherian ring (because $\mathbb Z$ is a PID and therefore Noetherian and if $R$ is Noetherian then so is $R[X]$). My task is to find a subring $R \subset \mathbb Z[X]$ with unity which would be non Noetherian.

I've already tried the subrings like $\mathbb Z[X^2]$ or $\mathbb Z[X^2, X^3]$, but unfortunately they are all Noetherian. Maybe all subrings are Noetherian? If this is the case, then how to prove it? If not, then what would be the counterexample?

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The subring $$R=\mathbb Z[2,2X,2X^2,2X^3,\dots]\subset \mathbb Z[X]$$ is not noetherian because its ideal $\langle 2,2X,2X^2,2X^3,\dots \rangle$ is not finitely generated.

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