Left Module Does Not Necessarily Induce Right Module over Ring

Theorem
Let $\struct {R, +_R, \times_R}$ be a 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 $\struct{G, +_G, \circ’}$ is not necessarily a right module over $\struct {R, +_R, \times_R}$

Proof
Proof by Counterexample

Let $\struct {S, +_S, \times_S}$ be a ring with zero 0.

Let $\struct {\map {\mathcal M_S} 2, +, \times}$ denote the ring of square matrices of order $2$ over $S$.

Let $R = \map {\mathcal M_S} 2$.

From Ring of Square Matrices over Ring is Ring, $\struct {R, +, \times}$ is a ring.

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

Also see

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