Definition:Ring (Abstract Algebra)/Addition

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The distributand $*$ of a ring $\struct {R, *, \circ}$ is referred to as ring addition, or just addition.

The conventional symbol for this operation is $+$, and thus a general ring is usually denoted $\struct {R, +, \circ}$.

Additive Group

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

Additive Inverse

Let $\struct {R, +, \circ}$ be a ring whose ring addition operation is $+$.

Let $a \in R$ be any arbitrary element of $R$.

The additive inverse of $a$ is its inverse under ring addition, denoted $-a$:

$a + \paren {-a} = 0_R$

where $0_R$ is the zero of $R$.