Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 2 / 9E (Groups, Rings and Modules)

A euclidean domain is an integral domain with a function $d$ on nonzero elements such that for $a,b\ne 0$ there are $q,r$ with $a=bq+r$ and either $r=0$ or $d(r)<d(b)$.

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.

The class $a$ is a unit modulo $p^e$ exactly when $\varphi (a)\ne 0$. This says a representative is not divisible by $p$, so it is coprime to $p^e$ and Bézout gives an inverse.

The class $a$ is nilpotent exactly when $\varphi (a)=0$. Divisibility by $p$ makes $a^e$ zero modulo $p^e$. Conversely, the image of a nilpotent in the field $R/(p)$ is nilpotent and therefore zero.

The Chinese remainder theorem identifiesThus a class is a unit exactly when its image modulo every $p_i$ is nonzero, equivalently when no $p_i$ divides a representative.

A unit modulo $d$ has nonzero image modulo every $p_i$. Any representative therefore satisfies the criterion in part (c), so it is already a unit modulo $I$. Its image is the given unit modulo $d$, proving surjectivity on unit groups.