Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2012 / 3 / a / Solution

A prime ideal is a proper ideal $I\ne R$ such that $ab\in I$ implies $a\in I$ or $b\in I$. A maximal ideal is a proper ideal $I\ne R$ such that there is no ideal strictly between $I$ and $R$.

Every ideal of the integers has the form $(n)=n\mathbb{Z}$ for a unique nonnegative integer $n$. Its prime ideals and maximal ideals are respectivelyThe zero ideal is a prime ideal because the integers form an integral domain. For a prime number $p$, divisibility of $ab$ by $p$ forces divisibility of one factor, so $(p)$ is a prime ideal. Also any ideal $(d)$ containing $(p)$ has $d\mid p$, so it equals $(p)$ or $(1)$; thus $(p)$ is a maximal ideal.

If $n>1$ is composite, write $n=ab$ with $1<a,b<n$. Then $ab\in (n)$ while neither factor belongs to $(n)$, so $(n)$ is not a prime ideal. A prime number $p\mid n$ gives $(n)⊊(p)⊊\mathbb{Z}$, so $(n)$ is not a maximal ideal either. The zero ideal is not maximal since $(0)⊊(2)⊊\mathbb{Z}$, and $(1)=\mathbb{Z}$ is not proper.