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[proof 1]
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 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 -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 not be homogeneous over the real numbers; said differently, there exist additive maps that are not of the form for some constant
In particular, there exist additive maps that are not linear maps.