# Definition:Additive Group of Ring

## Definition

Let $\struct {R, +, \circ}$ be a ring.

The group $\struct {R, +}$ is known as the **additive group of $R$**.

## Also defined as

Some sources make special issue of the nature of a group when its underlying set is a subset of, or derived directly from, numbers themselves.

In such treatments, a group whose operation is addition is then referred to as an **additive group**.

On $\mathsf{Pr} \infty \mathsf{fWiki}$ we consider **all** groups, whatever their nature, to be instances of the same abstract concept, and therefore make no such distinction.

Some sources confuse and muddy the water still further by calling an **additive group** *any* group whose notation is such that it uses $+$ as the symbol to denote the group operation and use $0$ to denote the identity.

## Also denoted as

Some sources write $\struct {R, +}$ as $R^+$ but this can be confused with the set of positive elements $R_+$ of an ordered ring.

## Also see

- Definition:Additive Group of Integers
- Definition:Additive Group of Integer Multiples
- Definition:Additive Group of Integers Modulo m

- Definition:Additive Group of Rational Numbers
- Definition:Additive Group of Real Numbers
- Definition:Additive Group of Complex Numbers

## Sources

- 1964: Iain T. Adamson:
*Introduction to Field Theory*... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 2$. Elementary Properties - 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): $\S 2.1$: Subrings: Notation $1$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**additive group** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**additive group**