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