Right Module over Ring Induces Left Module over Opposite Ring
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Theorem
Let $\struct {R, +_R, \times_R}$ be a ring.
Let $\struct {R, +_R, *_R}$ be the opposite ring of $\struct {R, +_R, \times_R}$.
Let $\struct{G, +_G, \circ}$ be a right module over $\struct {R, +_R, \times_R}$.
Let $\circ' : R \times G \to G$ be the binary operation defined by:
- $\forall \lambda \in R: \forall x \in G: \lambda \circ' x = x \circ \lambda$
Then $\struct {G, +_G, \circ'}$ is a left module over $\struct {R, +_R, *_R}$.
Proof
It is shown that $\struct {G, +_G, \circ'}$ satisfies the left module axioms.
By definition of the opposite ring:
- $\forall x, y \in S: x *_R y = y \times_R x$.
Left Module Axiom $\text M 1$: (Left) Distributivity over Module Addition
Let $\lambda \in R$ and $x, y \in G$.
\(\ds \lambda \circ' \paren{x +_G y}\) | \(=\) | \(\ds \paren {x +_G y} \circ \lambda\) | Definition of $\circ’$ | |||||||||||
\(\ds \) | \(=\) | \(\ds x \circ \lambda +_G y \circ \lambda\) | Right Module Axiom $\text {RM} 1$: (Right) Distributivity over Module Addition on $\struct{G, +_G, \circ}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \lambda \circ' x +_G \lambda \circ' y\) | Definition of $\circ’$ |
$\Box$
Left Module Axiom $\text M 2$: (Right) Distributivity over Scalar Addition
Let $\lambda, \mu \in R$ and $x \in G$.
\(\ds \paren {\lambda +_R \mu} \circ' x\) | \(=\) | \(\ds x \circ \paren {\lambda +_R \mu}\) | Definition of $\circ’$ | |||||||||||
\(\ds \) | \(=\) | \(\ds x \circ \lambda +_G x \circ \mu\) | Right Module Axiom $\text {RM} 2$: (Left) Distributivity over Scalar Addition on $\struct{G, +_G, \circ}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \lambda \circ' x +_G \mu \circ' x\) | Definition of $\circ’$ |
$\Box$
Left Module Axiom $\text M 3$: Associativity
Let $\lambda, \mu \in R$ and $x \in G$.
\(\ds \paren {\lambda *_R \mu} \circ' x\) | \(=\) | \(\ds x \circ \paren {\lambda *_R \mu}\) | Definition of $\circ’$ | |||||||||||
\(\ds \) | \(=\) | \(\ds x \circ \paren {\mu \times_R \lambda}\) | Definition of $*_R$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {x \circ \mu} \circ \lambda\) | Right Module Axiom $\text {RM} 3$: Associativity on $\struct {G, +_G, \circ}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\mu \circ' x} \circ \lambda\) | Definition of $\circ’$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \lambda \circ' \paren {\mu \circ' x}\) | Definition of $\circ'$ |
$\blacksquare$
Also see
Sources
- 2003: P.M. Cohn: Basic Algebra: Groups, Rings and Fields ... (previous) ... (next): Chapter $4$: Rings and Modules: $\S 4.1$: The Definitions Recalled