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{{Short description|Z-module homomorphism}}
{{About||additive functions in number theory|Additive function|additive functions on the reals|Cauchy's functional equation}}

In [[Abstract algebra|algebra]], an '''additive map''', '''<math>Z</math>-linear map''' or '''additive function''' is a [[Function (mathematics)|function]] <math>f</math> that preserves the addition operation:{{refn|{{citation|author1=[[Leslie Hogben]]|title=Handbook of Linear Algebra|publisher=CRC Press|year=2013|edition=3|isbn=9781498785600|pages=30–8}}}}
<math display=block>f(x + y) = f(x) + f(y)</math>
for every pair of elements <math>x</math> and <math>y</math> in the [[Domain of a function|domain]] of <math>f.</math> For example, any [[linear map]] is additive. When the domain is the [[real number]]s, this is [[Cauchy's functional equation]]. For a specific case of this definition, see [[additive polynomial]].

More formally, an additive map is a <math>\Z</math>-[[module homomorphism]]. Since an [[abelian group]] is a <math>\Z</math>-[[Module (mathematics)|module]], it may be defined as a [[group homomorphism]] between abelian groups.

A map <math>V \times W \to X</math> that is additive in each of two arguments separately is called a '''bi-additive map''' or a '''<math>\Z</math>-bilinear map'''.{{refn|{{citation|author=N. Bourbaki|title=Algebra Chapters 1–3|year=1989|publisher=Springer|page=243}}}}

== Examples ==

Typical examples include maps between [[Ring (mathematics)|ring]]s, [[vector space]]s, or [[Module (mathematics)|module]]s 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 <math>f</math> and <math>g</math> are additive maps, then the map <math>f + g</math> (defined [[pointwise]]) is additive.

== Properties ==

'''Definition of scalar multiplication by an integer'''

Suppose that <math>X</math> is an additive group with identity element <math>0</math> and that the inverse of <math>x \in X</math> is denoted by <math>-x.</math> For any <math>x \in X</math> and integer <math>n \in \Z,</math> let:
<math display=block>n x := \left\{
\begin{alignat}{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{alignat}
\right.</math>
Thus <math>(-1) x = - x</math> and it can be shown that for all integers <math>m, n \in \Z</math> and all <math>x \in X,</math> <math>(m + n) x = m x + n x</math> and <math>- (n x) = (-n) x = n (-x).</math>
This definition of scalar multiplication makes the cyclic subgroup <math>\Z x</math> of <math>X</math> into a [[Module (mathematics)|left <math>\Z</math>-module]]; if <math>X</math> is commutative, then it also makes <math>X</math> into a left <math>\Z</math>-module.

'''Homogeneity over the integers'''

If <math>f : X \to Y</math> is an additive map between additive groups then <math>f(0) = 0</math> and for all <math>x \in X,</math> <math>f(-x) = - f(x)</math> (where negation denotes the additive inverse) and<ref group=proof><math>f(0) = f(0 + 0) = f(0) + f(0)</math> so adding <math>-f(0)</math> to both sides proves that <math>f(0) = 0.</math> If <math>x \in X</math> then <math>0 = f(0) = f(x + (-x)) = f(x) + f(-x)</math> so that <math>f(-x) = - f(x)</math> where by definition, <math>(-1) f(x) := - f(x).</math> Induction shows that if <math>n \in \N</math> is positive then <math>f(n x) = n f(x)</math> and that the additive inverse of <math>n f(x)</math> is <math>n (- f(x)),</math> which implies that <math>f((-n) x) = f(n (-x)) = n f(-x) = n (- f(x)) = -(n f(x)) = (-n) f(x)</math> (this shows that <math>f(n x) = n f(x)</math> holds for <math>n < 0</math>). <math>\blacksquare</math></ref>
<math display=block>f(n x) = n f(x) \quad \text{ for all } n \in \Z.</math>
Consequently, <math>f(x - y) = f(x) - f(y)</math> for all <math>x, y \in X</math> (where by definition, <math>x - y := x + (-y)</math>).

In other words, every additive map is [[Homogeneous function|homogeneous over the integers]]. Consequently, every additive map between [[abelian group]]s is a [[Module homomorphism|homomorphism of <math>\Z</math>-modules]].

'''Homomorphism of <math>\Q</math>-modules'''

If the additive [[abelian group]]s <math>X</math> and <math>Y</math> are also a [[Unital module|unital]] [[Module (mathematics)|module]]s over the rationals <math>\Q</math> (such as real or complex [[vector space]]s) then an additive map <math>f : X \to Y</math> satisfies:<ref group=proof>Let <math>x \in X</math> and <math>q = \frac{m}{n} \in \Q</math> where <math>m, n \in \Z</math> and <math>n > 0.</math> Let <math>y := \frac{1}{n} x.</math> Then <math>n y = n \left(\frac{1}{n} x\right) = \left(n \frac{1}{n}\right) x = (1) x = x,</math> which implies <math>f(x) = f(n y) = n f(y) = n f\left(\frac{1}{n} x\right)</math> so that multiplying both sides by <math>\frac{1}{n}</math> proves that <math>f\left(\frac{1}{n} x\right) = \frac{1}{n} f(x).</math> Consequently, <math>f(q x) = f\left(\frac{m}{n} x\right) = m f\left(\frac{1}{n} x\right) = m\left(\frac{1}{n} f(x)\right) = q f(x).</math> <math>\blacksquare</math></ref>
<math display=block>f(q x) = q f(x) \quad \text{ for all } q \in \Q \text{ and } x \in X.</math>
In other words, every additive map is [[homogeneous over the rational numbers]]. Consequently, every additive maps between unital [[Module (mathematics)|<math>\Q</math>-modules]] is a [[Module homomorphism|homomorphism of <math>\Q</math>-modules]].

Despite being homogeneous over <math>\Q,</math> as described in the article on [[Cauchy's functional equation]], even when <math>X = Y = \R,</math> it is nevertheless still possible for the additive function <math>f : \R \to \R</math> to {{em|not}} be [[Real homogeneity|homogeneous over the real numbers]]; said differently, there exist additive maps <math>f : \R \to \R</math> that are {{em|not}} of the form <math>f(x) = s_0 x</math> for some constant <math>s_0 \in \R.</math>
In particular, there exist additive maps that are not [[linear map]]s.

== See also ==

* {{annotated link|Antilinear map}}

== Notes ==

{{reflist}}
{{reflist|group=note}}

'''Proofs'''

{{reflist|group=proof}}

== References ==

* {{citation|author1=[[Roger C. Lyndon]]|author2=[[Paul E. Schupp]]|title=Combinatorial Group Theory|publisher=Springer|year=2001}}

[[Category:Ring theory]]
[[Category:Morphisms]]
[[Category:Additive functions| ]]
[[Category:Types of functions]]

Additive map - Revision history

imported>Mgkrupa: added Category:Types of functions using HotCat