In algebra, an additive map, ${\displaystyle Z}$-linear map or additive function is a function ${\displaystyle f}$ that preserves the addition operation:^[1]

$${\displaystyle f(x+y)=f(x)+f(y)}$$

${\displaystyle x}$

${\displaystyle y}$

${\displaystyle f.}$

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]

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

Properties

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.}$$

${\displaystyle (-1)x=-x}$

${\displaystyle m,n\in \mathbb {Z} }$

${\displaystyle x\in X,}$

${\displaystyle (m+n)x=mx+nx}$

${\displaystyle -(nx)=(-n)x=n(-x).}$

${\displaystyle \mathbb {Z} x}$

${\displaystyle X}$

left ${\displaystyle \mathbb {Z} }$-module

${\displaystyle X}$

${\displaystyle X}$

${\displaystyle \mathbb {Z} }$

Homogeneity over the integers

If ${\displaystyle f:X\to Y}$ is an additive map between additive groups then ${\displaystyle f(0)=0}$ and for all ${\displaystyle x\in X,}$ ${\displaystyle f(-x)=-f(x)}$ (where negation denotes the additive inverse) and^{[proof 1]}

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

${\displaystyle f(x-y)=f(x)-f(y)}$

${\displaystyle x,y\in X}$

${\displaystyle x-y:=x+(-y)}$

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

Homomorphism of ${\displaystyle \mathbb {Q} }$-modules

If the additive abelian groups ${\displaystyle X}$ and ${\displaystyle Y}$ are also a unital modules over the rationals ${\displaystyle \mathbb {Q} }$ (such as real or complex vector spaces) then an additive map ${\displaystyle f:X\to Y}$ satisfies:^{[proof 2]}

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

${\displaystyle \mathbb {Q} }$-modules

homomorphism of ${\displaystyle \mathbb {Q} }$-modules

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