Definition:Additive Group of Ring

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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