Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 2 / 2G / a / Solution

Let $S=\{1,x,x^2,\dots \}$. Since a principal ideal domain is an integral domain and $x\ne 0$, $S$ is a multiplicative subset avoiding zero. The stated equivalence relation and operations are precisely those of the localization of a ring $S^{-1}R$.

For completeness, reflexivity and symmetry are immediate, while transitivity follows by multiplying the two cross-multiplication identities and cancelling a nonzero power of $x$. If $(r,x^n)\sim (s,x^m)$ and $(r^{\prime },x^{n^{\prime }})\sim (s^{\prime },x^{m^{\prime }})$, cross multiplication shows that the proposed sums and products are equivalent. Thus the operations do not depend on representatives. The usual fraction calculations then prove associativity, commutativity and distributivity, with identities $[(0,1)]$ and $[(1,1)]$. Therefore $R^{\prime }$ is a well-defined commutative ring.