Additive map

In algebra, an additive map, ${\displaystyle Z}$-linear map or additive function is a function ${\displaystyle f}$ that preserves the addition operation:^[1] $${\displaystyle f(x+y)=f(x)+f(y)}$$ for every pair of elements ${\displaystyle x}$ and ${\displaystyle y}$ in the domain of ${\displaystyle f.}$ 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 ${\displaystyle \mathbb{\mathbb{Z}}}$-module homomorphism. Since an abelian group is a ${\displaystyle \mathbb{\mathbb{Z}}}$-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 ${\displaystyle \mathbb{\mathbb{Z}}}$-bilinear map.^[2]

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.

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 ${\displaystyle n\in \mathbb{\mathbb{Z}},}$ let: $${\displaystyle nx:=\{\begin{array}{rlrlrlrlrlrlll}& & & 0& & & & & & & & & & \text{ when }n=0,\\ & & & x& & +\cdots +& & x& & \text{(}n& & \text{ summands) }& & \text{ when }n>0,\\ & (-& & x)& & +\cdots +(-& & x)& & \text{(}|n|& & \text{ summands) }& & \text{ when }n<0,\\ \end{array}}$$ Thus ${\displaystyle (-1)x=-x}$ and it can be shown that for all integers ${\displaystyle m,n\in \mathbb{\mathbb{Z}}}$ 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 ${\displaystyle \mathbb{\mathbb{Z}}x}$ of ${\displaystyle X}$ into a left ${\displaystyle \mathbb{\mathbb{Z}}}$-module; if ${\displaystyle X}$ is commutative, then it also makes ${\displaystyle X}$ into a left ${\displaystyle \mathbb{\mathbb{Z}}}$-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^{[proof 1]} $${\displaystyle f(nx)=nf(x)\text{ for all }n\in \mathbb{\mathbb{Z}}.}$$ 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 ${\displaystyle \mathbb{\mathbb{Z}}}$-modules.

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

If the additive abelian groups ${\displaystyle X}$ and ${\displaystyle Y}$ are also a unital modules over the rationals ${\displaystyle \mathbb{\mathbb{Q}}}$ (such as real or complex vector spaces) then an additive map ${\displaystyle f:X\to Y}$ satisfies:^{[proof 2]} $${\displaystyle f(qx)=qf(x)\text{ for all }q\in \mathbb{\mathbb{Q}}\text{ and }x\in X.}$$ In other words, every additive map is homogeneous over the rational numbers. Consequently, every additive maps between unital ${\displaystyle \mathbb{\mathbb{Q}}}$-modules is a homomorphism of ${\displaystyle \mathbb{\mathbb{Q}}}$-modules.

Despite being homogeneous over ${\displaystyle \mathbb{\mathbb{Q}},}$ 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.

• Antilinear map – Conjugate homogeneous additive map

Proofs

• Roger C. Lyndon; Paul E. Schupp (2001), Combinatorial Group Theory, Springer 

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