Definition:Additive Group of Ring

Definition
Let $\left({R, *, \circ}\right)$ be a ring.

The group $\left({R, *}\right)$ is known as the additive group of $R$.

Addition
The distributand $*$ of a ring $\left({R, *, \circ}\right)$ is referred to as ring addition, or just addition.

The conventional symbol for this operation is $+$, and a general ring is frequently denoted $\left({R, +, \circ}\right)$.

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

Additive Group is Abelian
Most treatments of this subject include as an axiom that the additive group of a ring is abelian.

This is true, and can be proved directly from the other axioms.