Past exam of the mathematics course of the University of Cambridge / 2017 / ib / Paper 3 / 11E / a

A Euclidean domain is an integral domain $R$ with a function $\nu :R\setminus \{0\}\to \mathbb{N}$ such that for each $a\in R$ and $b\ne 0$ there are $q,r\in R$ with $a=qb+r$ and either $r=0$ or $\nu (r)<\nu (b)$. Some conventions add a multiplicative monotonicity condition; it is unnecessary for this implication.

Let $I$ be a nonzero ideal. Choose $d\in I\setminus \{0\}$ minimizing $\nu (d)$, possible because the values lie in the nonnegative integers. For $a\in I$, division by $d$ gives $a=qd+r$, with $r=a-qd\in I$. A nonzero remainder would contradict minimality, so $r=0$ and $a\in (d)$. Since $d\in I$, the reverse inclusion also holds. The zero ideal is $(0)$. Thus