Additive map

Short description: Z-module homomorphism

For additive functions in number theory, see Additive function. For additive functions on the reals, see Cauchy's functional equation.

In algebra, an additive map, Z-linear map or additive function is a function f that preserves the addition operation:[1] f(x+y)=f(x)+f(y) for every pair of elements x and y in the domain of 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 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×W->X that is additive in each of two arguments separately is called a bi-additive map or a ℤ-bilinear map.[2]

## Contents

* 1 Examples
* 2 Properties
* 3 See also
* 4 Notes
* 5 References

## 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 f and g are additive maps, then the map f+g (defined pointwise) is additive.

## Properties

Definition of scalar multiplication by an integer

Suppose that X is an additive group with identity element 0 and that the inverse of x∈X is denoted by −x. For any x∈X and integer n∈ℤ, let: nx:={0 when n=0,x+⋯+x(n summands) when n>0,(−x)+⋯+(−x)(|n| summands) when n<0, Thus (−1)x=−x and it can be shown that for all integers m,n∈ℤ and all x∈X, (m+n)x=mx+nx and −(nx)=(−n)x=n(−x). This definition of scalar multiplication makes the cyclic subgroup ℤx 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->Y is an additive map between additive groups then f(0)=0 and for all x∈X, f(−x)=−f(x) (where negation denotes the additive inverse) and[proof 1] f(nx)=nf(x) for all n∈ℤ. Consequently, f(x−y)=f(x)−f(y) for all x,y∈X (where, by definition, 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 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->Y satisfies:[proof 2] f(qx)=qf(x) for all q∈ℚ and x∈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 ℚ, as described in the article on Cauchy's functional equation, even when X=Y=ℝ, it is nevertheless still possible for the additive function f:ℝ->ℝ to not be homogeneous over the real numbers; said differently, there exist additive maps f:ℝ->ℝ that are not of the form f(x)=s0x for some constant s0∈ℝ. In particular, there exist additive maps that are not linear maps.

## See also

* Antilinear map - Conjugate homogeneous additive map

## Notes

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. ↑ f(0)=f(0+0)=f(0)+f(0) so adding −f(0) to both sides proves that f(0)=0. If x∈X then 0=f(0)=f(x+(−x))=f(x)+f(−x) so that f(−x)=−f(x) where, by definition, (−1)f(x):=−f(x). Induction shows that if n∈ℕ is positive then f(nx)=nf(x) and that the additive inverse of nf(x) is n(−f(x)), which implies that f((−n)x)=f(n(−x))=nf(−x)=n(−f(x))=−(nf(x))=(−n)f(x) (this shows that f(nx)=nf(x) holds for n<0). ◼
2. ↑ Let x∈X and q=mn∈ℚ where m,n∈ℤ and n>0. Let y:=1nx. Then ny=n(1nx)=(n1n)x=(1)x=x, which implies f(x)=f(ny)=nf(y)=nf(1nx) so that multiplying both sides by 1n proves that f(1nx)=1nf(x). Consequently, f(qx)=f(mnx)=mf(1nx)=m(1nf(x))=qf(x). ◼

## References

* Roger C. Lyndon; Paul E. Schupp (2001), Combinatorial Group Theory, Springer

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