TITLE: Unital homomorphism
QUESTION [9 upvotes]: What is a unital homomorphism? Why are they important?

REPLY [7 votes]: 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 $R \to S$.