Right Module over Ring Induces Left Module over Opposite Ring

Theorem
Let $R = \struct {S, +_S, \times_S}$ be a ring.

Let $R^O = \struct {S, +_S, *_S}$ be the opposite ring.

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

Let $\circ’ : S \times G \to S$ be the binary operation defined by:
 * $\forall \lambda \in S: \forall x \in G: \lambda \circ’ x = x \circ \lambda $

Then $\struct{G, +_G, \circ’}$ is a left module over $R^O$.

Proof
It is shown that $\struct{G, +_G, \circ’}$ satisfies the left module axioms

By definition of the opposite ring:
 * $\forall x, y \in S: x *_S y = y \times_S x$.

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

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

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

Also see

 * Leigh.Samphier/Sandbox/Left Module over Ring Induces Right Module over Opposite Ring