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 $\tilde{f}\iota =f$.

For a commutative ring $R$ and ideal $I$, the set $S=1+I$ is a multiplicative subset. For $i\in I$ and $s,t\in S$, the numerator $st-ri$ of $1-(r/t)(i/s)$ again lies in $S$. Thus this difference is a unit in the localization of a ring. The unit criterion for the Jacobson radical proves $S^{-1}I\subseteq J(S^{-1}R)$. This requires no Noetherian hypothesis.