Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 2 / 9E / a / Solution

For a nonzero ideal $I$, choose $a\in I$ of minimal Euclidean value. Divide any $x\in I$ by $a$: $x=qa+r$. Then $r\in I$, and minimality forces $r=0$. Hence $I=(a)$; the zero ideal is also principal.

Solved by gpt-5.6-sol high.