Show polynomials $I$ is not finitely generated as $R$-module

Question

Let $R=\{a_0+a_1X+\cdots+a_nX^n\;|\;a_0\in\mathbb{Z},a_1,a_2,\dots ,a_n\in\mathbb{Q}, n\in\mathbb{Z}_{\geq 0}\}$ and $I=\{a_1X+\cdots+a_nX^n\;|\;a_1,a_2,\dots ,a_n\in\mathbb{Q}, n\in\mathbb{Z}^+\}$.

How to see $I$ is not finitely generated as an $R$-module?

Answer 1

Put more succinctly:

Given $R=\Bbb Z+X\Bbb Q[X]$ and $M=X\Bbb Q[X]$, why is $M$ is an infinitely-generated $R$-module?

Show the $X$-coefficients of a putative generating set for $M$ would generate $\Bbb Q$ as a $\Bbb Z$-module.

(The intuition here is that "mod $X^2$" we see that $R$ acts on $M$ just like $\Bbb Z$ acts on $\Bbb Q X$.)