Английская Википедия:Additive map
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In algebra, an additive map, <math>Z</math>-linear map or additive function is a function <math>f</math> that preserves the addition operation:Шаблон:Refn <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 <math>f.</math> 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.Шаблон:Refn
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 <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 && && &&EducationBot (обсуждение) 05:27, 1 января 2024 (+04) && &&~\text{ when } n = 0, \\ & &&x &&+ \cdots + &&x &&EducationBot (обсуждение) 05:27, 1 января 2024 (+04) \text{(} n &&\text{ summands) } &&~\text{ when } n > 0, \\ & (-&&x) &&+ \cdots + (-&&x) &&EducationBot (обсуждение) 05:27, 1 января 2024 (+04) \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 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[proof 1] <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 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:[proof 2] <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 <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 Шаблон:Em be homogeneous over the real numbers; said differently, there exist additive maps <math>f : \R \to \R</math> that are Шаблон:Em 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 maps.
See also
Notes
Proofs
References
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