Euclidean domain ⇒ PID

Euclidean domain ⇒ PID: If $R$ is a Euclidean domain , then $R$ is a principal ideal domain : every ideal of $R$ is a principal ideal .

Proof sketch: Let $I\neq (0)$ be an ideal and choose $a\in I$ of minimal Euclidean norm. For any $b\in I$, divide $b=qa+r$ with either $r=0$ or $\delta(r)<\delta(a)$. Since $r=b-qa\in I$, minimality forces $r=0$, hence $b\in (a)$. Therefore $I=(a)$. The argument uses the division property encoded by the Euclidean algorithm .