# Unital homomorphism

A unital homomorphism in between rings R and also S is a ring homomorphism that sends out the identification component of R to the identification component of S.

Homomorphisms (in between things in any kind of algebraic group like teams, rings, vector rooms, etc) maintain the algebraic framework, and also if you desire a map in between rings with an identification component, it is all-natural to need this to maintain this component (given that it pleases one-of-a-kind buildings).

A great deal of outcomes concerning rings simply will not function or else : as an example, a unital homomorphism of rings sends out devices to devices. A nonunital homomorphism does not need to do that. Nonunital homomorphisms can be really degenerate, as an example the absolutely no homomorphism.

An additional factor you desire homomorphisms to maintain the device is that this is just how you get a map $\operatorname{Spec S} \to \operatorname{Spec} R$ from a ring - homomorphism $R \to S$.

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