additive group
Noun
Meaning
(mathematics) Any group, or algebraic object that can be regarded as a group, whose binary operation is called addition and denoted +; specifically:
(group theory) Any (often, but not necessarily, abelian) group whose binary operation is called addition.
(algebra) Any of certain algebraic objects (ring, field or vector space, etc., whose definition includes a commutative operation called addition) regarded as a group under addition.
(category theory) Any group object (such as a group functor, group scheme, etc.), whose binary operation is called addition.
Source: en.wiktionary.orgExamples
Additive groups of rings The additive group of a ring is an abelian group, but not all abelian groups are additive groups of rings (with nontrivial multiplication).
Although the definition assumes that the additive group is abelian, this can be inferred from the other ring axioms.
Every infinite cyclic group is isomorphic to the additive group of Z, the integers.
For example, the ring with as additive group and trivial multiplication (i. e. for all ) has no maximal ideal (and of course no 1): Its ideals are precisely the additive subgroups.
Repeating in category language: is a morphism between the category of preferences with uncertainty and the category of reals as an additive group.
Similarly, the additive group Z of integers is not simple; the set of even integers is a non-trivial proper normal subgroup.