Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 3 / 10G / ii

Suppose $p$ is irreducible andThen $p=ab$ for some $b\in R$. Irreducibility says that either $a$ is a unit, in which case $(a)=R$, or $b$ is a unit, in which case $(a)=(p)$. Thus no proper ideal lies strictly between $(p)$ and $R$, so