Let $I$ be an ideal of the ring $\Bbb Z^n$. We know that $I$ is finitely generated as an ideal (finite product of noetherian is noetherian), but actually it is generated by a single element, as a finite product of PIDs is a PIR.
How do I find a generating element of $I$? Is there any algorithm? For instance how would you proceed for $I=\langle(4,0,2),(2,-2,0)\rangle$ or $J=\langle(-2,4,0,2),(2,-2,0,1)\rangle$?
Obviously, $\langle x_1,\dots,x_m\rangle=\langle x_1 \rangle+ \dots +\langle x_m \rangle$,but this is not always equal to $\langle x_1 + \cdots + x_m \rangle$ (see $m=2,x_2=-x_1$). So I'm not sure what to do.
Thank you!