Additive map

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

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: $$nx:=\{\begin{array}{cccccccccccccc}& & & 0& & & & & & & & & & \text{ when }n=0,\\ & & & x& & +\cdots +& & x& & \text{(}n& & \text{ summands) }& & \text{ when }n>0,\\ & (-& & x)& & +\cdots +(-& & x)& & \text{(}|n|& & \text{ summands) }& & \text{ when }n<0,\\ \end{array}$$ Thus $(-1)x=-x$ 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 $$f(nx)=nf(x)\text{ for all }n\in \mathbb{Z}.$$ 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: $$f(qx)=qf(x)\text{ for all }q\in \mathbb{Q}\text{ and }x\in X.$$ 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 $X=Y=ℝ,$ it is nevertheless still possible for the additive function $f:ℝ\to ℝ$ to Template:Em be homogeneous over the real numbers; said differently, there exist additive maps $f:ℝ\to ℝ$ that are Template:Em of the form $f(x)=s_{0}x$ for some constant $s_0\in ℝ.$ In particular, there exist additive maps that are not linear maps.

See also

• Template:Annotated link

Notes

Proofs

References

• Template:Citation