# Opposite Ring of Opposite Ring

## Theorem

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

Let $\struct {R, +, *}$ be the opposite ring of $\struct {R, +, \times}$.

Let $\struct {R, +, \circ}$ be the opposite ring of $\struct {R, +, *}$.

Then $\struct {R, +, \circ} = \struct {R, +, \times}$.

## Proof

By definition of the opposite ring:

$\forall x, y \in S: x * y = y \times x$
$\forall x, y \in S: x \circ y = y * x$

Hence for all $x,y \in S$:

$x \circ y = y * x = x \times y$

The result follows.

$\blacksquare$