Opposite Ring is Ring
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Theorem
Let $\struct {R, +, \times}$ be a ring.
Let $\struct {R, +, *}$ be the opposite ring of $\struct {R, +, \times}$.
Then $\struct {R, +, *}$ is a ring.
Proof
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.
$\blacksquare$
Sources
- 2003: P.M. Cohn: Basic Algebra: Groups, Rings and Fields ... (previous) ... (next): Chapter $4$: Rings and Modules: $\S 4.1$: The Definitions Recalled