In
algebra
Algebra () is one of the areas of mathematics, 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 mathem ...
, 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 measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
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 com ...
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 whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
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 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 com ...
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 com ...
s
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 whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
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
Roger Conant Lyndon (December 18, 1917 – June 8, 1988) was an American mathematician, for many years a professor at the University of Michigan.. He is known for Lyndon words, the Curtis–Hedlund–Lyndon theorem, Craig–Lyndon interpolation a ...
, author2=
Paul E. Schupp, title=Combinatorial Group Theory, publisher=Springer, year=2001
Ring theory
Morphisms