Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 4 / 9E / a / ii / Solution

The annihilator $I={\mathrm{Ann}}_R(M)$ is an ideal. For $r\notin I$, choose $m$ with $rm\ne 0$. Irreducibility and part (i) give $s\in R$ such that $srm=m$. The element $(sr-1)$ therefore kills the generator $m$, and hence all of $M$; thus $sr-1\in I$. Every nonzero class in $R/I$ consequently has an inverse, so $R/I$ is a field.