Additive map
# Additive map

In algebra, an additive map, $Z$-linear map or additive function is a function $f$ that preserves the addition operation:

$f(x+y)=f(x)+f(y)$

for every pair of elements $x$ and $y$ in the domain of $f.$ 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 $\mathbb {Z}$-module homomorphism. Since an abelian group is a $\mathbb {Z}$-module, it may be defined as a group homomorphism between abelian groups.

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

## Examples

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

## Properties

Definition of scalar multiplication by an integer

Suppose that $X$ is an additive group with identity element $0$ and that the inverse of $x\in X$ is denoted by $-x.$ For any $x\in X$ and integer $n\in \mathbb {Z} ,$ let:

$nx:=\left\{{\begin{alignedat}{9}&&&0&&&&&&~~~~&&&&~{\text{ when }}n=0,\\&&&x&&+\cdots +&&x&&~~~~{\text{}n&&{\text{ summands) }}&&~{\text{ when }}n>0,\\&(-&&x)&&+\cdots +(-&&x)&&~~~~{\text{}|n|&&{\text{ summands) }}&&~{\text{ when }}n<0,\\\end{alignedat}}\right.$

Thus $(-1)x=-x$ and it can be shown that for all integers $m,n\in \mathbb {Z}$ and all $x\in X,$ $(m+n)x=mx+nx$ and $-(nx)=(-n)x=n(-x).$ This definition of scalar multiplication makes the cyclic subgroup $\mathbb {Z} x$ of $X$ into a left $\mathbb {Z}$-module; if $X$ is commutative, then it also makes $X$ into a left $\mathbb {Z}$-module.

Homogeneity over the integers

If $f:X\to Y$ is an additive map between additive groups then $f(0)=0$ and for all $x\in X,$ $f(-x)=-f(x)$ (where negation denotes the additive inverse) and

$f(nx)=nf(x)\quad {\text{ for all }}n\in \mathbb {Z} .$

Consequently, $f(x-y)=f(x)-f(y)$ for all $x,y\in X$ (where by definition, $x-y:=x+(-y)$).

In other words, every additive map is homogeneous over the integers. Consequently, every additive map between abelian groups is a homomorphism of $\mathbb {Z}$-modules.

Homomorphism of $\mathbb {Q}$-modules

If the additive abelian groups $X$ and $Y$ are also a unital modules over the rationals $\mathbb {Q}$ (such as real or complex vector spaces) then an additive map $f:X\to Y$ satisfies:

$f(qx)=qf(x)\quad {\text{ for all }}q\in \mathbb {Q} {\text{ and }}x\in X.$

In other words, every additive map is homogeneous over the rational numbers. Consequently, every additive maps between unital $\mathbb {Q}$-modules is a homomorphism of $\mathbb {Q}$-modules.

Despite being homogeneous over $\mathbb {Q} ,$ as described in the article on Cauchy's functional equation, even when $X=Y=\mathbb {R} ,$ it is nevertheless still possible for the additive function $f:\mathbb {R} \to \mathbb {R}$ to not be homogeneous over the real numbers; said differently, there exist additive maps $f:\mathbb {R} \to \mathbb {R}$ that are not of the form $f(x)=s_{0}x$ for some constant $s_{0}\in \mathbb {R} .$ In particular, there exist additive maps that are not linear maps.

