Solution
= Solution
The annihilator $I=\operatorname{Ann}_R(M)$ is an <ideal>. For $r\notin I$, choose $m$ with $rm\ne0$. 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.