In algebra, an additive map, -linear map or additive function is a function that preserves the addition operation:[1] for every pair of elements and in the domain of 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 -module homomorphism. Since an abelian group is a -module, it may be defined as a group homomorphism between abelian groups.

A map that is additive in each of two arguments separately is called a bi-additive map or a -bilinear map.[2]

Examples

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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   and   are additive maps, then the map   (defined pointwise) is additive.

Properties

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Definition of scalar multiplication by an integer

Suppose that   is an additive group with identity element   and that the inverse of   is denoted by   For any   and integer   let:   Thus   and it can be shown that for all integers   and all     and   This definition of scalar multiplication makes the cyclic subgroup   of   into a left  -module; if   is commutative, then it also makes   into a left  -module.

Homogeneity over the integers

If   is an additive map between additive groups then   and for all     (where negation denotes the additive inverse) and[proof 1]   Consequently,   for all   (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  -modules.

Homomorphism of  -modules

If the additive abelian groups   and   are also a unital modules over the rationals   (such as real or complex vector spaces) then an additive map   satisfies:[proof 2]   In other words, every additive map is homogeneous over the rational numbers. Consequently, every additive maps between unital  -modules is a homomorphism of  -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   to not be homogeneous over the real numbers; said differently, there exist additive maps   that are not of the form   for some constant   In particular, there exist additive maps that are not linear maps.

See also

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Notes

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  1. ^ Leslie Hogben (2013), Handbook of Linear Algebra (3 ed.), CRC Press, pp. 30–8, ISBN 9781498785600
  2. ^ N. Bourbaki (1989), Algebra Chapters 1–3, Springer, p. 243

Proofs

  1. ^   so adding   to both sides proves that   If   then   so that   where by definition,   Induction shows that if   is positive then   and that the additive inverse of   is   which implies that   (this shows that   holds for  ).  
  2. ^ Let   and   where   and   Let   Then   which implies   so that multiplying both sides by   proves that   Consequently,    

References

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