Ring Negative is Unique

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
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.