In algebra, an additive map, ${\displaystyle Z}$-linear map or additive function is a function ${\displaystyle f}$ that preserves the addition operation: $${\displaystyle f(x+y)=f(x)+f(y)}$$ 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.

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 ${\displaystyle f}$ and ${\displaystyle g}$ are additive maps, then the map ${\displaystyle f+g}$ (defined pointwise) is additive.

Definition of scalar multiplication by an integer

Suppose that ${\displaystyle X}$ is an additive group with identity element ${\displaystyle 0}$ and that the inverse of ${\displaystyle x\in X}$ is denoted by ${\displaystyle -x.}$ For any ${\displaystyle x\in X}$ and integer ${\displaystyle n\in \mathbb {Z} ,}$ let: $${\displaystyle nx:=\left\{{\begin{alignedat}{9}&&&0&&&&&&~~~~&&&&~{\text{ when }}n=0,\\&&&x&&+\cdots +&&x&&~~~~{\text{(}}n&&{\text{ summands) }}&&~{\text{ when }}n>0,\\&(-&&x)&&+\cdots +(-&&x)&&~~~~{\text{(}}|n|&&{\text{ summands) }}&&~{\text{ when }}n<0,\\\end{alignedat}}\right.}$$ Thus ${\displaystyle (-1)x=-x}$ and it can be shown that for all integers ${\displaystyle m,n\in \mathbb {Z} }$ and all ${\displaystyle x\in X,}$ ${\displaystyle (m+n)x=mx+nx}$ and ${\displaystyle -(nx)=(-n)x=n(-x).}$ This definition of scalar multiplication makes the cyclic subgroup ${\displaystyle \mathbb {Z} x}$ of ${\displaystyle X}$ into a left ${\displaystyle \mathbb {Z} }$-module; if ${\displaystyle X}$ is commutative, then it also makes ${\displaystyle X}$ into a left ${\displaystyle \mathbb {Z} }$-module.

Homogeneity over the integers