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# Ring homomorphism

In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. If R and S are rings and f : R -> S is a function, we require
• f(a + b) = f(a) + f(b) for all a and b in R
• f(ab) = f(a) f(b) for all a and b in R
• f(1) = 1

### Properties

Directly from these definitions, one can deduce:

• f(0) = 0
• f(-a) = -f(a)
• If a has a multiplicative inverse in R, then f(a) has a multiplicative inverse in S and we have f(a-1) = (f(a))-1. Therefore, f induces a group homomorphism from the group of units of R to the group of units of S.
• The kernel of f, defined as ker(f) = {a in R : f(a) = 0} is an ideal in R. Every ideal in R arises from some ring homomorphism in this way. f is injective if and only if the ker(f) = {0}.
• If f is bijective, then its inverse f -1 is also a ring homomorphism. f is called an isomorphism in this case, and the rings R and S are called isomorphic. From the standpoint of ring theory, isomorphic rings cannot be distinguished.
• If Rp is the smallest subring contained in R and Sp is the smallest subring contained in S, then every ring homomorphism f : R -> S induces a ring homomorphism fp : Rp -> Sp. This can sometimes be used to show that between certain rings R and S, no ring homomorphisms R -> S can exist.
• The composition of two ring homomorphisms is a ring homomorphism; the class of all rings together with the ring homomorphisms forms a category.