Additive map

For additive functions in number theory, see Additive function. For additive functions on the reals, see Cauchy's functional equation.

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