Abstract algebra/Ring Homomorphisms

Definition

Let R and S be two rings. Then a function $f:R\to S$ is called a ring homomorphism if for every $r_1,r_2\in R$, the following properties hold:

$f(r_1{r}_2)=f(r_1)f(r_2),$

$f(r_1+r_2)=f(r_1)+f(r_2).$

In other words, f is a ring homomorphism if it preserves additive and multiplicative structure.

Furthermore, if R and S are rings with unity and $f(1_R)=1_S$, then f is called a unital ring homomorphism.

Relationship between homomorphisms and ideals

Rings, like groups, have factor objects that are kernels of homomorphisms. Let $f:R\to S$ be a ring homomorphism. Let us determine the structure of the kernel of f.

If a and b are in the kernel of f, i.e. $f(a)=f(b)=0$, and r is any element of R, then

$f(a+b)=f(a)+f(b)=0$,

$f(a-b)=f(a)-f(b)=0$,

$f(ar)=f(a)f(r)=0$.

Therefore $\ker (f)$ is an ideal of R. Furthermore, we have an isomorphism theorem for rings analogous to the one for groups:

$R/\ker (f)\cong \text{im}(f)$.