Additive map
Z-module homomorphism / From Wikipedia, the free encyclopedia
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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, -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]