# Definition:Induced Homomorphism between Localizations of Ring

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## Definition

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.