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Definition: A ring homomorphism phi: R -> R' is a map from one ring to another which is compatible with the laws of composition and which carries the unit element 1 of R to the unit element 1 in R' - a map such that, for all a and b in R, phi(a+b)=phi(a)+phi(b), phi(ab)=phi(a)phi(b), and phi(1)=1.
Source: Algebra, second edition by Michael Artin