In
algebra, 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 of
For example, any
linear map 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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
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 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.
Examples
Typical examples include maps between
rings,
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
s, 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
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 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 space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
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 maps.
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
Types of functions