Left Module over Commutative Ring induces Bimodule

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

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

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

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

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

 * Right Module over Commutative Ring induces Bimodule