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 R_{p} is the smallest subring contained in R and S_{p} is the smallest subring contained in S, then every ring homomorphism f : R -> S induces a ring homomorphism f_{p} : R_{p} -> S_{p}. 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.
Examples