\begin{definition}[Definition:Induced Homomorphism between Localizations of Ring]
Let $A$ be a commutative ring with unity.
Let $S, T \subseteq A$ be multiplicatively closed subsets.
Let $S$ be a subset of the saturation of $T$.
The '''induced homomorphism''' between localizations $A_S \to A_T$ is the unique $A$-algebra homomorphism between them.
\end{definition}