Is it true every ideal of $\mathbb{Z}[x]$ is finitely generated?

Question

Let $I$ be any ideal of $\mathbb {Z}[x]$, and $I=\left<{f_1,f_2 , ...}\right>$.

Without loss of generality, $f_1$ has minimal (absolute value) coefficient of constant term (if there exist).

Then, let $g$ be any element of $I$ .

We have for some integer $m$ that $g - mf_1$ is element of $I$ (minimal coefficient and division algorithm).

I guess this process terminate finitely, but I don't know how can I do next step.

( ex. $g-mf_1, g-mf_1-nf_2, ...$)

Please help me and thanks for in advance.

Answer 1

$\mathbb Z$ is Noetherian, and by Hilbert's basis theorem, so is $\mathbb Z[x].$ Therefore, by definition, all its ideals are finitely generated.