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.

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 Field

• 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 $1$