# Additive map

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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 [algebra](/source/Abstract_algebra), an '''additive map''', '''<math>\Z</math>-linear map''' or '''additive function''' is a [function](/source/Function_(mathematics)) <math>f</math> that preserves the addition operation:{{refn|{{citation|author1=[Leslie Hogben](/source/Leslie_Hogben)|title=Handbook of Linear Algebra|publisher=CRC Press|year=2013|edition=3|isbn=9781498785600|page=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](/source/Domain_of_a_function) of {{tmath| f }}. For example, any [linear map](/source/linear_map) is additive. When the domain is the [real number](/source/real_number)s, this is [Cauchy's functional equation](/source/Cauchy's_functional_equation). For a specific case of this definition, see [additive polynomial](/source/additive_polynomial).

More formally, an additive map is a <math>\Z</math>-[module homomorphism](/source/module_homomorphism).  Since an [abelian group](/source/abelian_group) is a <math>\Z</math>-[module](/source/Module_(mathematics)), it may be defined as a [group homomorphism](/source/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 |author-link=Nicolas Bourbaki |title=Algebra Chapters 1–3 |year=1989 |publisher=Springer |page=243 }}}}

== Examples ==

Typical examples include maps between [ring](/source/Ring_(mathematics))s, [vector space](/source/vector_space)s, or [module](/source/Module_(mathematics))s that preserve the [additive group](/source/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](/source/pointwise)) is additive.

== Properties ==

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

Suppose that <math>X</math> is an additive group with [identity element](/source/identity_element) <math>0</math> and that the inverse of <math>x \in X</math> is denoted by {{tmath| -x }}. For any <math>x \in X</math> and integer {{tmath| n \in \Z }}, 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 {{tmath| x \in X }}, <math>(m + n) x = m x + n x</math> and {{tmath|1= - (n x) = (-n) x = n (-x) }}. 
This definition of [scalar multiplication](/source/scalar_multiplication) makes the cyclic subgroup <math>\Z x</math> of <math>X</math> into a [left <math>\Z</math>-module](/source/Module_(mathematics)); 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 {{tmath| x \in X }}, <math>f(-x) = - f(x)</math> (where negation denotes the [additive inverse](/source/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 {{tmath|1= f(0) = 0 }}. 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, {{tmath|1= (-1) f(x) := - f(x) }}. 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 {{tmath| n (-f(x)) }}, 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 {{tmath| n < 0 }}). <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, {{tmath|1= x - y := x + (-y) }}). 

In other words, every additive map is [homogeneous over the integers](/source/Homogeneous_function). Consequently, every additive map between [abelian group](/source/abelian_group)s is a [homomorphism of <math>\Z</math>-modules](/source/Module_homomorphism). 

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

If the additive [abelian group](/source/abelian_group)s <math>X</math> and <math>Y</math> are also a [unital](/source/Unital_module) [module](/source/Module_(mathematics))s over the rationals <math>\Q</math> (such as real or complex [vector space](/source/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 {{tmath| n > 0 }}. Let {{tmath|1= y := \frac{1}{n} x }}. Then {{tmath|1= n y = n \left(\frac{1}{n} x\right) = \left(n \frac{1}{n}\right) x = (1) x = x}}, 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 {{tmath|1= f\left(\frac{1}{n} x\right) = \frac{1}{n} f(x) }}. Consequently, {{tmath|1= 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>\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](/source/homogeneous_over_the_rational_numbers). Consequently, every additive maps between unital [<math>\Q</math>-modules](/source/Module_(mathematics)) is a [homomorphism of <math>\Q</math>-modules](/source/Module_homomorphism). 

Despite being homogeneous over {{tmath| \Q }}, as described in the article on [Cauchy's functional equation](/source/Cauchy's_functional_equation), even when {{tmath|1= X = Y = \R }}, it is nevertheless still possible for the additive function <math>f : \R \to \R</math> to {{em|not}} be [homogeneous over the real numbers](/source/Real_homogeneity); 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 {{tmath| s_0 \in \R }}. 
In particular, there exist additive maps that are not [linear map](/source/linear_map)s with respect to an existing ring structure of the [codomain](/source/codomain).

== See also ==

* {{annotated link|Antilinear map}}

== Notes ==

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

'''Proofs'''

{{reflist|group=proof}}

== References ==

* {{citation |author1=[Roger C. Lyndon](/source/Roger_C._Lyndon) |author2=[Paul E. Schupp](/source/Paul_E._Schupp) |title=Combinatorial Group Theory |publisher=Springer |year=2001 }}

Category:Ring theory
Category:Morphisms
Category:Additive functions
Category:Types of functions

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Adapted from the Wikipedia article [Additive map](https://en.wikipedia.org/wiki/Additive_map) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Additive_map?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.