# Non Noetherian subring of Z[X]

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?

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.