Z-module homomorphism
In algebra, an additive map, -linear map or additive function is a function that preserves the addition operation:[1]
for every pair of elements
and
in the
domain of
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 -module homomorphism. Since an abelian group is a -module, it may be defined as a group homomorphism between abelian groups.
A map that is additive in each of two arguments separately is called a bi-additive map or a -bilinear map.[2]
- ^ Leslie Hogben (2013), Handbook of Linear Algebra (3 ed.), CRC Press, pp. 30–8, ISBN 9781498785600
- ^ N. Bourbaki (1989), Algebra Chapters 1–3, Springer, p. 243