Product with Ring Negative

Theorem
Let $\left({R, +, \circ}\right)$ be a ring.

Then:
 * $\forall x, y \in \left({R, +, \circ}\right) : \left({-x}\right) \circ y = - \left({x \circ y}\right) = x \circ \left({-y}\right)$

where $\left({-x}\right)$ denotes the negative of $x$.

Proof
We have:

So from group axiom $G3$ as applied to $\left({R, +}\right)$:
 * $\left({-x}\right) \circ y = -\left({x \circ y}\right)$

The proof that $x \circ \left({-y}\right) = - \left({x \circ y}\right)$ follows identical lines.

Also see

 * Product of Ring Negatives