Additive map

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: $${\displaystyle nx:=\{\begin{array}{cccccccccccccc}& & & 0& & & & & & & & & & \text{ when }n=0\\ & & & x& & +\cdots +& & x& & \text{(}n& & \text{ summands) }& & \text{ when }n>0\\ & (-& & x)& & +\cdots +(-& & x)& & \text{(}|n|& & \text{ summands) }& & \text{ when }n<0.\end{array}}$$ 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

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 $${\displaystyle f(nx)=nf(x)\text{ for all }n\in \mathbb{Z}.}$$ Consequently, ${\displaystyle f(x-y)=f(x)-f(y)}$ for all ${\displaystyle x,y\in X}$ (where, by definition, ⁠${\displaystyle x-y:=x+(-y)}$⁠).

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: $${\displaystyle f(qx)=qf(x)\text{ for all }q\in \mathbb{Q}\text{ and }x\in X.}$$ In other words, every additive map is homogeneous over the rational numbers. Consequently, every additive maps between unital ${\displaystyle \mathbb{Q}}$-modules is a 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 with respect to an existing ring structure of the codomain.