Product of Ring Negatives

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Theorem

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


Then:

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

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


Proof

We have:

\(\displaystyle \paren {-x} \circ \paren {-y}\) \(=\) \(\displaystyle -\paren {x \circ \paren {-y} }\) Product with Ring Negative
\(\displaystyle \) \(=\) \(\displaystyle -\paren {-\paren {x \circ y} }\) Product with Ring Negative
\(\displaystyle \) \(=\) \(\displaystyle x \circ y\) Negative of Ring Negative

$\blacksquare$


Also see


Sources