# Product with Ring Negative/Corollary

## Corollary to Product with Ring Negative

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

Then:

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

## Proof

 $\displaystyle \paren {-1_R} \circ x$ $=$ $\displaystyle -\paren {1_R \circ x}$ Product with Ring Negative $\displaystyle$ $=$ $\displaystyle -x$ Definition of Unity of Ring

$\blacksquare$