Left Module induces Right Module over same Ring iff Actions are Commutative
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Theorem
Let $\struct {R, +_R, \times_R}$ be a ring.
Let $\struct {G, +_G, \circ}$ be a left module over $\struct {R, +_R, \times_R}$.
Let $\circ': G \times R \to G$ be the binary operation defined by:
- $\forall \lambda \in R: \forall x \in G: x \circ' \lambda = \lambda \circ x$
Then $\struct {G, +_G, \circ'}$ is a right module over $\struct {R, +_R, \times_R}$ if and only if:
- $\forall \lambda, \mu \in R: \forall x \in G: \paren {\lambda \times_R \mu} \circ x = \paren {\mu \times_R \lambda} \circ x$
Proof
Necessary Condition
Let $\struct {G, +_G, \circ'}$ be a right module over $\struct {R, +_R, \times_R}$.
Then:
| \(\ds \paren {\lambda \times_R \mu} \circ x\) | \(=\) | \(\ds x \circ' \paren {\lambda \times_R \mu}\) | Definition of $\circ'$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {x \circ' \lambda} \circ' \mu\) | Right Module Axiom $\text {RM} 3$: Associativity | |||||||||||
| \(\ds \) | \(=\) | \(\ds \mu \circ \paren {\lambda \circ x}\) | Definition of $\circ'$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\mu \times_R \lambda} \circ x\) | Left Module Axiom $\text M 3$: Associativity |
$\Box$
Sufficient Condition
Let the scalar multiplication $\circ$ satisfy:
- $\forall \lambda, \mu \in R: \forall x \in G: \paren {\lambda \times_R \mu} \circ x = \paren {\mu \times_R \lambda} \circ x$
It needs to be shown that $\struct {G, +_G, \circ'}$ satisfies the right module axioms.
Right Module Axiom $\text {RM} 1$: (Right) Distributivity over Module Addition
Let $\lambda, \mu \in R, x \in G$.
Then:
| \(\ds \paren {x +_G y} \circ' \lambda\) | \(=\) | \(\ds \lambda \circ \paren {x +_G y}\) | Definition of $\circ'$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \lambda \circ x +_G \lambda \circ y\) | Left Module Axiom $\text M 1$: (Left) Distributivity over Module Addition | |||||||||||
| \(\ds \) | \(=\) | \(\ds x \circ' \lambda +_G y \circ' \lambda\) | Definition of $\circ'$ |
$\Box$
Right Module Axiom $\text {RM} 2$: (Left) Distributivity over Scalar Addition
Let $\lambda \in R, x, y \in G$.
Then:
| \(\ds x \circ' \paren{\lambda +_R \mu}\) | \(=\) | \(\ds \paren {\lambda +_R \mu} \circ x\) | Definition of $\circ'$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \lambda \circ x +_G \mu \circ y\) | Left Module Axiom $\text M 2$: (Right) Distributivity over Scalar Addition | |||||||||||
| \(\ds \) | \(=\) | \(\ds x \circ' \lambda +_G x \circ' \mu\) | Definition of $\circ'$ |
$\Box$
Right Module Axiom $\text {RM} 3$: Associativity
Let $\lambda, \mu \in R, x \in G$.
Then:
| \(\ds x \circ' \paren {\lambda \times_R \mu}\) | \(=\) | \(\ds \paren {\lambda \times_R \mu} \circ x\) | Definition of $\circ'$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\mu \times_R \lambda} \circ x\) | Assumption | |||||||||||
| \(\ds \) | \(=\) | \(\ds \mu \circ \paren {\lambda \circ x}\) | Left Module Axiom $\text M 3$: Associativity | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {x \circ' \lambda} \circ' \mu\) | Definition of $\circ'$ |
$\blacksquare$