Finding generators for an ideal of $\Bbb{Z}[x]$ We know that $\Bbb{Z}$ is Noetherian. Hence, we can conclude that $\Bbb{Z}[x]$ is Noetherian, too. Consider the ideal generated by $\langle 2x^2+2,3x^3+3,5x^5+5,…,px^p+p,…\rangle$ for all prime natural numbers $p$. How can I determine a finite number of elements which generate this ideal? Clearly the Euclidean algorithm is not valid in $\Bbb{Z}[x]$.
Thank you

$$ (1 - 11 x - x^2 + 5 x^3 - 5 x^4) (2 x^2 + 2) + 4 (1 + x) (3 x^3 + 3) - 
 2 (1 - x) (5 x^5 + 5) = 4$$
$$-112 (1 - x^3 - x^5) (5 x^5 + 5) - 80 (x + x^3) (7 x^7 + 7) + \\
 187 (1 - x^3 + x^6 - x^9) (3 x^3 + 3) + 51 x (11 x^{11} + 11) = x+1$$
Conversely,
$2x^2+2 = 2(x+1)(x-1)+4$, $px^p+p = p(x+1)(1-x+\dots + x^{p-1})$ for $p$ odd.
