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 $x$.

Corollary
Let $\left({R, +, \circ}\right)$ be a ring with unity $1_R$.

Then:
 * $\forall x \in R: \left({-1_R}\right) \circ x = -x$

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: