 Universal property of localization

The localization map $\iota :R\to S^{-1}R$ sends every element of $S$ to a unit. If a ring homomorphism $f:R\to T$ also sends every element of $S$ to a unit, there is a unique homomorphism $\tilde{f}:S^{-1}R\to T$ with $\tilde{f}(r/s)=f(r)f(s{)}^{-1}$ and $\tilde{f}\iota =f$.