Negative of Ring Negative
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Theorem
Let $\struct {R, +, \circ}$ be a ring.
Let $a \in R$ and let $-a$ be the ring negative of $a$.
Then:
- $-\paren {-a} = a$
Proof
The ring negative is, by definition of a ring, the inverse element of $a$ in the additive group $\struct {R, +}$.
The result then follows from Inverse of Group Inverse.
$\blacksquare$
Sources
- 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 2$. Elementary Properties
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Integral Domains: $\S 4$. Elementary Properties: Theorem $2 \ \text{(i)}$