Left Module over Commutative Ring induces Right Module

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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 $\circ’ : G \times R \to G$ 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 a right module over $\struct {R, +_R, \times_R}$.

Proof

From Ring is Commutative iff Opposite Ring is Itself, $\struct {R, +_R, \times_R}$ is its own opposite ring.

From Left Module over Ring Induces Right Module over Opposite Ring, $\struct{G, +_G, \circ’}$ is a right module over $\struct {R, +_R, \times_R}$.

$\blacksquare$

Also see

Sources