# 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 $x$.

### Corollary

Let $\struct {R, +, \circ}$ be a ring with unity $1_R$.

Then:

$\forall x \in R: \paren {-1_R} \circ x = -x$

## Proof

We have:

 $\displaystyle \paren {x + \paren {-x} } \circ y$ $=$ $\displaystyle 0_R \circ y$ Definition of Ring Zero $\displaystyle$ $=$ $\displaystyle 0_R$ Ring Product with Zero $\displaystyle \leadstoandfrom \ \$ $\displaystyle \paren {x \circ y} + \paren {\paren {-x} \circ y}$ $=$ $\displaystyle 0_R$ Ring Axiom $D$: $\circ$ is distributive over $+$

So from Group Axiom $G \, 3$: Inverses 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.

$\blacksquare$