In algebra, an additive map, -linear map or additive function is a function that preserves the addition operation: in the domain of . 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.

-module homomorphism. Since an abelian group-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 -bilinear map.

Examples

Typical examples include maps between ringss, vector spaces, or moduless 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.

Properties

Definition of scalar multiplication by an integer

is an additive group with identity element is denoted by and integer and all and . This definition of scalar multiplication into a \Z-module-module.

Homogeneity over the integers

and for all (where negation denotes the additive inverse) and (where, by definition, ).

In other words, every additive map is homogeneous over the integers. Consequently, every additive map between abelian groups is a \Z-modules.

-modules

If the additive abelian groups are also a unital modules (such as real or complex vector spaces satisfies: In other words, every additive map is homogeneous over the rational numbers. Consequently, every additive maps between unital \Q-modules is a \Q-modules.

Despite being homogeneous over , as described in the article on Cauchy's functional equation, even when to be homogeneous over the real numbers that are for some constant . In particular, there exist additive maps that are not linear maps with respect to an existing ring structure of the codomain.

See also

Notes

Proofs

References