Opposite Ring of Opposite Ring
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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.
$\blacksquare$