An additive group is a
group of which the group operation is to be thought of as ''addition'' in some sense. It is usually
abelian, and typically written using the symbol + for its binary operation.
This terminology is widely used with structures equipped with several operations for specifying the structure obtained by forgetting the other operations. Examples include the ''additive group'' of the
integers
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
, of a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
and of a
ring. This is particularly useful with rings and
fields to distinguish the additive underlying group from the
multiplicative group
In mathematics and group theory, the term multiplicative group refers to one of the following concepts:
*the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referre ...
of the
invertible element
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...
s.
In older terminology, an additive subgroup of a ring has also been known as a ''modul'' or ''module'' (not to be confused with a
module).
References
Algebraic structures
Group theory
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