Noun
(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.orgAdditive 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). Source: Internet
Although the definition assumes that the additive group is abelian, this can be inferred from the other ring axioms. Source: Internet
Every infinite cyclic group is isomorphic to the additive group of Z, the integers. Source: Internet
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. Source: Internet
Repeating in category language: is a morphism between the category of preferences with uncertainty and the category of reals as an additive group. Source: Internet
Similarly, the additive group Z of integers is not simple; the set of even integers is a non-trivial proper normal subgroup. Source: Internet