= Every Euclidean domain is a principal ideal domain
{title2=$\text{Euclidean domain}\Longrightarrow\text{principal ideal domain}$}
Let $R$ be a <Euclidean domain> with <Euclidean function> $\nu$, and let $I$ be a nonzero <ideal>. Choose $b\in I\setminus\{0\}$ minimizing the nonnegative <integer> $\nu(b)$. For $a\in I$, divide $a=qb+r$ with $r=0$ or $\nu(r)<\nu(b)$. Since $r=a-qb\in I$, the latter case contradicts minimality. Hence every $a$ is a multiple of $b$, giving $I=(b)$. The zero <ideal> is $(0)$, so every ideal is a <principal ideal> and $R$ is a <principal ideal domain>.
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