Additive map

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

In algebra, an '''additive map''', '''<math>\Z</math>-linear map''' or '''additive function''' is a 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|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 of {{tmath| 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 <math>\Z</math>-module homomorphism. Since an abelian group is a <math>\Z</math>-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 |author-link=Nicolas Bourbaki |title=Algebra Chapters 1–3 |year=1989 |publisher=Springer |page=243 }}}}

== 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 <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 {{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 makes the cyclic subgroup <math>\Z x</math> of <math>X</math> into a 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 {{tmath| x \in X }}, <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 {{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. Consequently, every additive map between abelian groups is a homomorphism of <math>\Z</math>-modules.

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

If the additive abelian groups <math>X</math> and <math>Y</math> are also a unital modules over the rationals <math>\Q</math> (such as real or complex vector spaces) 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. Consequently, every additive maps between unital <math>\Q</math>-modules is a homomorphism of <math>\Q</math>-modules.

Despite being homogeneous over {{tmath| \Q }}, as described in the article on 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; 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 maps with respect to an existing ring structure of the codomain.

== See also ==

* {{annotated link|Antilinear map}}

== Notes ==

'''Proofs'''

== 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