Definition:Additive Inverse

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Definition

Additive Inverse in Ring

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 $R$.


Additive Inverse in Field

Let $\struct {F, +, \times}$ be a field whose addition operation is $+$.

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


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

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

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


Additive Inverse of Number

The concept is often encountered in the context of numbers:

Let $\Bbb F$ be one of the standard number systems: $\N$, $\Z$, $\Q$, $\R$, $\C$.

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


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

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


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, the inverses of a group whose operation is addition are then referred to as additive inverses.

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 inverse an inverse in 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 see

  • Results about additive inverses can be found here.


Sources

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