Commutative Ring - Ring Homomorphisms

Ring Homomorphisms

As usual in algebra, a function f between two objects that respects the structures of the objects in question is called homomorphism. In the case of rings, a ring homomorphism is a map f : R → S such that

f(a + b) = f(a) + f(b), f(ab) = f(a)f(b) and f(1) = 1.

These conditions ensure f(0) = 0, but the requirement that the multiplicative identity element 1 is preserved under f would not follow from the two remaining properties. In such a situation S is also called an R-algebra, by understanding that s in S may be multiplied by some r of R, by setting

r · s := f(r) · s.

The kernel and image of f are defined by ker (f) = {r ∈ R, f(r) = 0} and im (f) = f(R) = {f(r), r ∈ R}. The kernel is an ideal of R, and the image is a subring of S.