Product with Ring Negative

Theorem
Let $\struct {R, +, \circ}$ be a ring.

Then:
 * $\forall x, y \in \struct {R, +, \circ}: \paren {-x} \circ y = -\paren {x \circ y} = x \circ \paren {-y}$

where $\paren {-x}$ denotes the negative of $x$.

Proof
We have:

So from as applied to $\struct {R, +}$:
 * $\paren {-x} \circ y = -\paren {x \circ y}$

The proof that $x \circ \paren {-y} = -\paren {x \circ y}$ follows identical lines.

Also see

 * Product of Ring Negatives