Product with Ring Negative

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

Then:


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


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

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:

$$ $$ $$