Prove that every ideal of $\Bbb{Z}$ has the form $m\Bbb{Z}=\{mk, k\in \Bbb{Z}\}$
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2026-02-22 22:55:15.1771800915
Prove that every ideal of $\Bbb{Z}$ has the form $m\Bbb{Z}=\{mk, k\in \Bbb{Z}\}$
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Let $I$ be an ideal of $\Bbb{Z}$ and let $m=\min\{i\in I;i\gt 0\}$. Consider $x\in I$. By the Euclidean division property of $\Bbb{Z}$, there exists $q\in\Bbb{Z}$, $r\in \Bbb{Z},0\leq r\lt i$ such that
$$x=iq+r$$
$i\in I$ implies $iq\in I$ and $x-iq=r\in I$. So $r=0$ otherwise we would have found an element $r$ of $I$ such that $0\lt r\lt i$ in contradiction with the definition of $i$.
So any element of $I$ can be written as $iq$ and so $I=i\Bbb{Z}$