Existence of Homomorphism between Localizations of Ring

Theorem
Let $A$ be a commutative ring with unity.

Let $S, T \subseteq A$ be multiplicatively closed subsets.


 * $(1): \quad$ There exists an $A$-algebra homomorphism $f : A_S \to A_T$ between localizations.
 * $(2): \quad S$ is a subset of the saturation of $T$.
 * $(3): \quad$ The saturation of $S$ is a subset of the saturation of $T$.
 * $(4): \quad$ Every prime ideal meeting $S$ also meets $T$.

Also see

 * Uniqueness of Homomorphism between Localizations of Ring