Additive map

In algebra an additive map, Z-linear map or additive function is a function that preserves the addition operation:

f(x+y)=f(x)+f(y).

for any two elements x and y in the domain. For example, any linear map is additive. When the domain is the real numbers, this is Cauchy's functional equation. For a specific case of this definition, see additive polynomial. Any homomorphism f between abelian groups is additive by this definition.

More formally, 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 $R_{2}$ .

An additive map is not required to preserve the product operation of the ring.

If $f$ and $g$ are additive maps, then the map $f+g$ (defined pointwise) is additive.

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