Additive Group

An additive group is a group where the operation is called addition and is denoted . In an additive group, the identity element is called zero, and the inverse of the element is denoted (minus ). The symbols and terminology are borrowed from the additive groups of numbers: the ring of integers , the field of rational numbers , the field of real numbers , and the field of complex numbers are all additive groups.

In general, every ring and every field is an additive group. An important class of examples is given by the polynomial rings with coefficients in a ring . In the additive group of the sum is performed by adding the coefficients of equal terms,

(1)

Modules, abstract vector spaces, and algebras are all additive groups.

The sum of vectors of the vector space is defined componentwise,

(2)

and so is the sum of matrices with entries in a ring ,

(3)

which is part of the -module structure of the set of matrices .

Any quotient group of an Abelian additive group is again an additive group with respect to the induced addition of cosets, defined by

(4)

for all .

This is the case for all the examples above as well as for with , 3, ..., where

(5)

which is the sum of the residue classes of and , sometimes denoted and . These are also examples of cyclic additive groups; is generated by the element , which means that

(6)

In any additive group , the integer multiples of every element can be considered for every integer ,

(7)

This multiplication by integers makes a -module iff is Abelian.

In abstractly defined groups, the additive notation is preferred when the operation is commutative. This is normally not the case for groups of maps; there, the composition is usually treated as a multiplication. One natural exception is the group of translations of -dimensional Euclidean space. If and are the translations determined by the vectors and of , then

(8)

which means that the composition is equivalent to the sum of translation vectors. The group of translations of the Euclidean space can therefore be identified with the additive group of vectors of .