Additive map

In algebra, an additive map, ${\displaystyle Z}$-linear map or additive function is a function ${\displaystyle f}$ that preserves the addition operation:^[1] $${\displaystyle f(x+y)=f(x)+f(y)}$$ for every pair of elements ${\displaystyle x}$ and ${\displaystyle y}$ in the domain of ${\displaystyle f.}$ 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 ${\displaystyle \mathbb {Z} }$-module homomorphism. Since an abelian group is a ${\displaystyle \mathbb {Z} }$-module, it may be defined as a group homomorphism between abelian groups.

A map ${\displaystyle V\times W\to X}$ that is additive in each of two arguments separately is called a bi-additive map or a ${\displaystyle \mathbb {Z} }$-bilinear map.^[2]