Right Module over Commutative Ring induces Bimodule

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

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 $\struct{G, +_G, \circ’, \circ}$ is a bimodule over $\struct {R, +_R, \times_R}$.

Proof
From Right Module over Commutative Ring induces Left Module, $\struct{G, +_G, \circ’}$ is a left module.

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

Then:

Hence $\struct{G, +_G, \circ’, \circ}$ is a bimodule over $\struct {R, +_R, \times_R}$ by definition.

Also see

 * Left Module over Commutative Ring induces Bimodule