Right Module over Ring Induces Left Module over Opposite Ring

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:
 * $\forall x, y \in S: x *_R y = y \times_R x$.

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

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

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

Also see

 * Left Module over Ring Induces Right Module over Opposite Ring