Opposite Ring is Ring

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

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

Then $\struct {R, +, *}$ is a ring.


By definition of the opposite ring:

$\forall x, y \in R: x * y = y \times x$.

By definition of the ring $R$, $\struct {R, +}$ is an abelian group.

To complete the proof, it remains to be shown that $\struct {R, *}$ is a semigroup.

That is, it remains to show that $\struct{R, *}$ is associative.

Let $a, b, c \in R$.

\(\ds a * \paren {b * c}\) \(=\) \(\ds \paren{b * c} \times a\) Definition of $*$
\(\ds \) \(=\) \(\ds \paren {c \times b} \times a\) Definition of $*$
\(\ds \) \(=\) \(\ds c \times \paren {b \times a}\) Ring Axiom $\text M1$: Associativity of Product
\(\ds \) \(=\) \(\ds c \times \paren {a * b}\) Definition of $*$
\(\ds \) \(=\) \(\ds \paren {a * b} * c\) Definition of $*$

The result follows.