Definition

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 $R$.

The concept is often encountered in the context of numbers:

Let $\Bbb F$ be one of the standard sets of numbers

Let $a \in \Bbb F$ be any arbitrary number.

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

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