Definition:Negative of Element

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Definition

Ring Negative

Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$.

Let $x \in R$.


The inverse of $x$ with respect to the addition operation $+$ in the additive group $\struct {R, +}$ of $R$ is referred to as the (ring) negative of $x$ and is denoted $-x$.


That is, the (ring) negative of $x$ is the element $-x$ of $R$ such that:

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


Field Negative

As a field is also a ring, the same definition can be used:


Let $\struct {F, +, \times}$ be a field whose zero is $0_F$.

Let $x \in F$.


The inverse of $x$ with respect to the addition operation $+$ in the additive group $\struct {F, +}$ of $F$ is referred to as the (field) negative of $x$ and is denoted $-x$.


That is, the (field) negative of $x$ is the element $-x$ of $F$ such that:

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