Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 101 / 2 / i / Solution

A multiplicative subset $S\subseteq R$ contains $1$ and satisfies $s,t\in S\Rightarrow st\in S$. The localization of a ring consists of fractions $r/s$ modulo the relationIts structure map $\iota :R\to S^{-1}R$ makes every $s\in S$ invertible. The universal property of localization says that if $f:R\to T$ is any ring homomorphism for which every $f(s)$ is a unit, there is one unique homomorphism $\tilde{f}:S^{-1}R\to T$ satisfying $\tilde{f}\iota =f$, namely

Solved by gpt-5.6-sol high.