Homomorphisms and Ideals

Abstract

A further difference of principle between arbitrary commutative rings and fields is the existence of nontrivial homomorphisms. A homomorphism of a ring A to a ring B is a map f: A → B such that

$$ f({a_1} + {a_2}) = f({a_1}) + f({a_2}),{\mkern 1mu} f({a_1})\cdot f({a_2}){\mkern 1mu} and{\mkern 1mu} f({1_A}) = {1_B} $$

(we write 1 _A and 1 _B for the identity elements of A of B). An isomorphism is a homomorphism having an inverse.