Editing Additive group (section)

{{Short description|Group with an addition as its operation}}
{{Wiktionary}}
An '''additive group''' is a [[group (mathematics)|group]] of which the group operation is to be thought of as ''addition'' in some sense.  It is usually [[abelian group|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''<ref>{{citation |first=N. |last=Bourbaki |title=Algebra I: Chapters 1–3 |chapter=§8.1 Rings |chapter-url=https://books.google.com/books?id=STS9aZ6F204C&pg=PA97 |year=1998 |publisher=Springer |isbn=978-3-540-64243-5 |page=97 |orig-year=1970}}</ref> of the [[integers]], of a [[vector space]] and of a [[ring (mathematics)|ring]]. This is particularly useful with rings and [[field (mathematics)|fields]] to distinguish the additive underlying group from the [[multiplicative group]] of the [[unit (ring theory)|invertible element]]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 (mathematics)|module]]).<ref>{{cite web |title=MathOverflow: The Origin(s) of Modular and Moduli |url=https://mathoverflow.net/questions/300013/the-origins-of-modular-and-moduli/300076#300076 |access-date=8 March 2024}}</ref>