which respects the operations of addition and multiplication. If
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.