Product with Ring Negative

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

Then:


 * $(1): \quad \forall x, y \in \left({R, +, \circ}\right) : \left({-x}\right) \circ y = - \left({x \circ y}\right) = x \circ \left({-y}\right)$


 * $(2): \quad \forall x, y \in \left({R, +, \circ}\right): \left({-x}\right) \circ \left({-y}\right) = x \circ y$

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

Proof
$\left({-x}\right) \circ y = - \left({x \circ y}\right)$:

We have:

So from the elementary consequences of the group axioms, $\left({-x}\right) \circ y = -\left({x \circ y}\right)$.

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

$\left({-x}\right) \circ \left({-y}\right) = x \circ y$:

We have: