Additive map

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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[edit]

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[edit]

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[edit]

Notes[edit]

  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[edit]