Additive map

In mathematics an additive map of ring $R_{1}$ into ring $R_{2}$ is a homomorphism

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

of the additive group of $R_{1}$ into the additive group of $R_{2}$.

According to the definition of a homomorphism of an additive group, an additive map $f$ of a ring $R_{1}$ into a ring $R_{2}$ satisfies

$f(a+b)=f(a)+f(b)\,.$

We do not expect that an additive map preserves the product operation of the ring.

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

Additive map of a division ring

Let $D$ be a division ring of characteristic $0$. We can represent an additive map $f:D\to D$ of the division ring $D$ as

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

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

References

• Leslie Hogben, Richard A. Brualdi, Anne Greenbaum, Roy Mathias, Handbook of linear algebra, CRC Press, 2007

• Roger C. Lyndon, Paul E. Schupp, Combinatorial Group Theory, Springer, 2001