Additive map

Homomorphism

$f:R_{1}\to R_{2}$

of additive group of ring $R_{1}$ into additive group of ring $R_{2}$ is called additive map of ring $R_{1}$ into ring $R_{2}$.

According to definition of homomorphism of additive group, additive map $f$ of ring $R_{1}$ into ring $R_{2}$ holds

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

We do not expect that additive map of ring holds product.

Since $f$ and $g$ are additive maps, then map $f+g$ is additive. Similarly, the map $afb$ is additive, when $a,b\in R_{2}$.

Additive map of division ring

Let $D$ be division ring of characteristic $0$. We can represent additive map

$f:D\to D$

of division ring $D$ as

$f(x)={}_{(s)0}f\ x\ {}_{(s)1}f$

We assume sum over index $s$. The number of items depends on the function $f$. Expressions ${}_{(s)0}f,{}_{(s)1}f\in D$ are called components of additive map.