Negative of Ring Negative

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 $2 \ \text{(i)}$
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