##
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^{[proof 1]}

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

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:^{[proof 2]}

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

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.