Additive map

In algebra, an additive 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 . For example, any linear map is additive. When the domain is the integers (ℤ), this is called a \Z -linear map. 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 \Z -module homomorphism. Since an abelian group is a \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 \Z -bilinear map.

In algebra, an additive 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 . For example, any linear map is additive. When the domain is the integers (ℤ), this is called a \Z -linear map. 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 \Z -module homomorphism. Since an abelian group is a \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 \Z -bilinear map. 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 Thus (-1) x = - x and it can be shown that for all integers m, n \in \Z and all , (m + n) x = m x + n x and . This definition of scalar multiplication makes the cyclic subgroup \Z x of X into a left \Z -module; if X is commutative, then it also makes X into a left \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 , f(-x) = - f(x) (where negation denotes the additive inverse) and f(n x) = n f(x) \quad \text{ for all } n \in \Z. Consequently, f(x - y) = f(x) - f(y) for all x, y \in X (where, by definition, ). In other words, every additive map is homogeneous over the integers. Consequently, every additive map between abelian groups is a homomorphism of \Z -modules. Homomorphism of \Q -modules If the additive abelian groups X and Y are also a unital modules over the rationals \Q (such as real or complex vector spaces) then an additive map f : X \to Y satisfies: f(q x) = q f(x) \quad \text{ for all } q \in \Q \text{ and } x \in X. In other words, every additive map is homogeneous over the rational numbers. Consequently, every additive maps between unital \Q -modules is a homomorphism of \Q -modules. Despite being homogeneous over , as described in the article on Cauchy's functional equation, even when , it is nevertheless still possible for the additive function f : \R \to \R to be homogeneous over the real numbers; said differently, there exist additive maps f : \R \to \R that are of the form f(x) = s_0 x for some constant . 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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