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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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A lot of results about rings just won't work otherwise: for instance, a unital homomorphism of rings sends units to units. A nonunital homomorphism doesn't have to do that. Nonunital homomorphisms can be very degenerate, e.g. the zero homomorphism.

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 $R \to S$.

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