In
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
, an additive map,
-linear map or additive function is a
function that preserves the addition operation:
for every pair of elements
and
in the
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
**Domain of holomorphy of a function
* ...
of
For example, any
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
is additive. When the domain is the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, 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
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
is a
-
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
, 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.
Examples
Typical examples include maps between
rings,
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s, or
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
s 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 In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined ...
) is additive.
Properties
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
[ 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 ). ]
Consequently,
for all
(where by definition,
).
In other words, every additive map is
homogeneous over the integers. Consequently, every additive map between
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s is a
homomorphism of -modules.
Homomorphism of
-modules
If the additive
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s
and
are also a
unital module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
s over the rationals
(such as real or complex
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s) then an additive map
satisfies:
[Let and where and Let Then which implies so that multiplying both sides by proves that Consequently, ]
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 be
homogeneous over the real numbers; said differently, there exist additive maps
that are of the form
for some constant
In particular, there exist additive maps that are not
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
s.
See also
*
Notes
Proofs
References
* {{citation, author1=
Roger C. Lyndon, author2=
Paul E. Schupp, title=Combinatorial Group Theory, publisher=Springer, year=2001
Ring theory
Morphisms