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Additive map

Z-module homomorphism

Wikipedia contributorsAdapted for Fathomly · View authors
2 min read
Source updated May 16, 2026

In algebra, an additive map, \mathbb {Z}-linear map or additive function is a function f that preserves the addition operation: 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.

01Examples

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.

02Properties

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 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: 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 with respect to an existing ring structure of the codomain.

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Sources and credits

This article is adapted from the Wikipedia article “Additive map”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Related topics

Antilinear map

In mathematics, a function f : V → W {\displaystyle f:V\to W} between two complex vector spaces is said to be antilinear or conjugate-linear if f = f ( x ) + f ( y ) (additivity) f ( s x ) = s ¯ f ( x ) (conjugate homogeneity) {\displaystyle {\begin{alignedat}{9}f(x+y)&=f(x)+f(y)&&\qquad {\text{ (additivity) }}\\f(sx)&={\overline {s}}f(x)&&\qquad {\text{ (conjugate homogeneity) }}\\\end{alignedat}}} hold for all vectors x , y ∈ V {\displaystyle x,y\in V} and every complex number s , {\displaystyle s,} where s ¯ {\displaystyle {\overline {s}}} denotes the complex conjugate of s . {\displaystyle s.} Antilinear maps stand in contrast to linear maps, which are additive maps that are homogeneous rather than conjugate homogeneous.