Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 20G / c

Given a nonzero fractional $I$, scale it to an integral ideal $aI$. By (i), $(aI)J=(r)$ for some nonzero ideal $J$. Then $I^{-1}=a{r}^{-1}J$ is fractional and $I{I}^{-1}=R$.

If every fractional ideal is invertible and $I\subset R$ is nonzero, choose $I^{-1}$ and clear denominators: $0\ne c{I}^{-1}\subset R$. Then $I(c{I}^{-1})=(c)$ is principal.