# Negative of Division Product

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## Theorem

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

Let $\struct {U_R, \circ}$ be the group of units of $\struct {R, +, \circ}$.

Then:

- $\displaystyle \forall x \in R: -\frac x z = \frac {-x} z = \frac x {-z}$

where $\dfrac x z$ is defined as $x \circ \paren {z^{-1} }$, that is the division product of $x$ by $z$.

## Proof

Follows directly from Product of Negative with Product Inverse and the definition of division product.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 23$: Theorem $23.7 \ (1)$