Additive group

{{Short description|Group with an addition as its operation}}

{{Wiktionary}}

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{{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}} of the integers, of a vector space and of a ring. This is particularly useful with rings and fields to distinguish the additive underlying group from the multiplicative group of the invertible elements.

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).{{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}}