Ring Negative is Unique
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Theorem
Let $\struct {R, +, \circ}$ be a ring.
Let $a \in R$.
Then the ring negative $-a$ of $a$ is unique.
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 in Group is Unique.
$\blacksquare$
Sources
- 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 2$. Elementary Properties