Is $ R =\{a_{2n} x^{2n} + ... + a_2 x^2 + a_0\mid n \in \Bbb N\}$, subring of $\Bbb Z[x]$, noetherian?

Question

Is $ R =\{a_{2n} x^{2n} + ... + a_2 x^2 + a_0\mid n \in \Bbb N\}$}, subring of $\Bbb Z[x]$, noetherian ?

Answer 1

Well, $R$ is isomorphic to $\mathbb{Z}[x]$ via the isomorphism

\begin{align*} \mathbb{Z}[x] &\to R \\ f(x) &\mapsto f(x^2). \end{align*}

In particular $R$ is noetherian.