# Unital homomorphism

What is a unital homomorphism? Why are they important?

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A unital homomorphism between rings R and S is a ring homomorphism that sends the identity element of R to the identity element of S.

Homomorphisms (between objects in any algebraic category like groups, rings, vector spaces, etc.) preserve the algebraic structure, and if you want a map between rings with an identity element, it is natural to require this to preserve this element (since it satisfies unique properties).

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Another reason you want homomorphisms to preserve the unit is that this is how you get a map $\operatorname{Spec S} \to \operatorname{Spec} R$ from a ring-homomorphism $R \to S$.