In algebra, an additive map,
that preserves the addition operation:[1]
for every pair of elements
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, it may be defined as a group homomorphism between abelian groups.
that is additive in each of two arguments separately is called a bi-additive map or a
[2] 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.
(defined pointwise) is additive.
Definition of scalar multiplication by an integer Suppose that
is an additive group with identity element
{\displaystyle nx:=\left\{{\begin{alignedat}{9}&&&0&&&&&&~~~~&&&&~{\text{ when }}n=0,\\&&&x&&+\cdots +&&x&&~~~~{\text{(}}n&&{\text{ summands) }}&&~{\text{ when }}n>0,\\&(-&&x)&&+\cdots +(-&&x)&&~~~~{\text{(}}|n|&&{\text{ summands) }}&&~{\text{ when }}n<0,\\\end{alignedat}}\right.}
and it can be shown that for all integers
This definition of scalar multiplication makes the cyclic subgroup
is commutative, then it also makes
(where negation denotes the additive inverse) and[proof 1]
In other words, every additive map is homogeneous over the integers.
Consequently, every additive map between abelian groups is a homomorphism of
-modules If the additive abelian groups
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
as described in the article on Cauchy's functional equation, even when
to not be homogeneous over the real numbers; said differently, there exist additive maps
In particular, there exist additive maps that are not linear maps.