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:

\(\displaystyle \paren{ \lambda \times_R \mu} \circ x\) \(=\) \(\displaystyle x \circ’ \paren{ \lambda \times_R \mu}\) Definition of $\circ’$
\(\displaystyle \) \(=\) \(\displaystyle \paren {x \circ’ \lambda} \circ’ \mu\) Right module axiom $(RM \, 3)$ : Associativity of Scalar Multiplication
\(\displaystyle \) \(=\) \(\displaystyle \mu \circ \paren {\lambda \circ x}\) Definition of $\circ’$
\(\displaystyle \) \(=\) \(\displaystyle \paren {\mu \times_R \lambda} \circ x\) Left module axiom $(M \, 3)$ : Associativity of Scalar Multiplication

$\Box$

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.


$(RM \, 1)$ : Scalar Multiplication (Right) Distributes over Module Addition

Let $\lambda, \mu \in R, x \in G$.

Then:

\(\displaystyle \paren {x +_G y} \circ’ \lambda\) \(=\) \(\displaystyle \lambda \circ \paren {x +_G y}\) Definition of $\circ’$
\(\displaystyle \) \(=\) \(\displaystyle \lambda \circ x +_G \lambda \circ y\) Left module axiom $(M \, 1)$ on $\circ$
\(\displaystyle \) \(=\) \(\displaystyle x \circ’ \lambda +_G y \circ’ \lambda\) Definition of $\circ’$

$\Box$

$(RM \, 2)$ : Scalar Multiplication (Left) Distributes over Scalar Addition

Let $\lambda \in R, x, y \in G$.

Then:

\(\displaystyle x \circ’ \paren{\lambda +_R \mu}\) \(=\) \(\displaystyle \paren{\lambda +_R \mu} \circ x\) Definition of $\circ’$
\(\displaystyle \) \(=\) \(\displaystyle \lambda \circ x +_G \mu \circ y\) Left module axiom $(M \, 2)$ on $\circ$
\(\displaystyle \) \(=\) \(\displaystyle x \circ’ \lambda +_G x \circ’ \mu\) Definition of $\circ’$

$\Box$

$(RM \, 3)$ : Associativity of Scalar Multiplication

Let $\lambda, \mu \in R, x \in G$.

Then:

\(\displaystyle x \circ’ \paren{\lambda \times_R \mu}\) \(=\) \(\displaystyle \paren{\lambda \times_R \mu} \circ x\) Definition of $\circ’$
\(\displaystyle \) \(=\) \(\displaystyle \paren{\mu \times_R \lambda} \circ x\) Assumption
\(\displaystyle \) \(=\) \(\displaystyle \mu \circ \paren{\lambda \circ x}\) Left module axiom $(M \, 3)$ on $\circ$
\(\displaystyle \) \(=\) \(\displaystyle \paren {x \circ’ \lambda} \circ’ \mu\) Definition of $\circ’$

$\blacksquare$

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