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 every pair of 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.

More formally, an additive map is a Z-module homomorphism. Since an abelian group is a Z-module, it may be defined as a group homomorphism between abelian groups.

Typical examples include maps between rings, vector spaces, or modules that preserve the additive group. An additive map does not necessarily preserve any other structure of the object, for example the product operation of a 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