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