Right Module induces Left Module over same Ring iff Actions are Commutative

Theorem
Let $\struct {R, +_R, \times_R}$ be a ring.

Let $\struct{G, +_G, \circ}$ be a right module over $\struct {R, +_R, \times_R}$.

Let $\circ’ : G \times R \to R$ 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 $\struct {R, +_R, \times_R}$ :
 * $\forall \lambda, \mu \in R: \forall x \in G: x \circ \paren{ \lambda \times \mu} = x \circ \paren{\mu \times \lambda}$

Necessary Condition
Let $\struct{G, +_G, \circ’}$ be a left module over $\struct {R, +_R, \times_R}$.

Then:

Sufficient Condition
Let the scalar multiplication $\circ$ satisfy:
 * $\forall \lambda, \mu \in R: \forall x \in G: x \circ \paren{ \lambda \times \mu} = x \circ \paren{\mu \times \lambda}$

It needs to be shown that $\struct{G, +_G, \circ’}$ satisfies the left module axioms.

$(M \, 1)$ Scalar Multiplication (Left) Distributes over Module Addition
Let $\lambda, \mu \in R, x \in G$.

Then:

$(M \, 2)$ Scalar Multiplication (Right) Distributes over Scalar Addition
Let $\lambda \in R, x, y \in G$.

Then:

$(M \, 3)$ Associativity of Scalar Multiplication
Let $\lambda, \mu \in R, x \in G$.

Then:

Also see

 * Leigh.Samphier/Sandbox/Left Module induces Right Module over same Ring iff Actions are Commutative