Ideal not finitely generated Let $R=\{a_0+a_1 X+a_2 X^2 +\cdots + a_n X^n\}$, where $a_0$ is an integer and the rest of the coefficients are rational numbers.
Let $I=\{a_1 X+a_2 X^2+\cdots +a_n X^n\}$ where all of the coefficients are rational numbers.
Prove that I is an ideal of R.
Show further that I is not finitely generated as an R-module.
I have managed to prove that I is an ideal of R, by showing that I is the kernel of the evaluation map that maps a polynomial in Q[x] to its constant term. Hence I is an ideal of R.
However, I am stuck at showing I is not finitely generated as an R-module.
Sincere thanks for any help.

Hint: $\frac{1}{p}X \not\in P_1R + P_2R + \dots + P_kR$, where $P_1, P_2,  \dots, P_k \in I$, with coefficients $\frac{a_1}{b_1}, \frac{a_2}{b_2}, \dots, \frac{a_k}{b_k}$ in degree one, and $p$ does not divide $b_1 b_2 \dots b_k$.
