Right Module over Ring Induces Left Module over Opposite Ring

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Theorem

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

Let $\struct {R, +_R, *_R}$ be the opposite ring of $\struct {R, +_R, \times_R}$.

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 $\struct {G, +_G, \circ'}$ is a left module over $\struct {R, +_R, *_R}$.


Proof

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

By definition of the opposite ring:

$\forall x, y \in S: x *_R y = y \times_R x$.


Left Module Axiom $\text M 1$: (Left) Distributivity over Module Addition

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

\(\ds \lambda \circ' \paren{x +_G y}\) \(=\) \(\ds \paren {x +_G y} \circ \lambda\) Definition of $\circ’$
\(\ds \) \(=\) \(\ds x \circ \lambda +_G y \circ \lambda\) Right Module Axiom $\text {RM} 1$: (Right) Distributivity over Module Addition on $\struct{G, +_G, \circ}$
\(\ds \) \(=\) \(\ds \lambda \circ' x +_G \lambda \circ' y\) Definition of $\circ’$

$\Box$


Left Module Axiom $\text M 2$: (Right) Distributivity over Scalar Addition

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

\(\ds \paren {\lambda +_R \mu} \circ' x\) \(=\) \(\ds x \circ \paren {\lambda +_R \mu}\) Definition of $\circ’$
\(\ds \) \(=\) \(\ds x \circ \lambda +_G x \circ \mu\) Right Module Axiom $\text {RM} 2$: (Left) Distributivity over Scalar Addition on $\struct{G, +_G, \circ}$
\(\ds \) \(=\) \(\ds \lambda \circ' x +_G \mu \circ' x\) Definition of $\circ’$

$\Box$


Left Module Axiom $\text M 3$: Associativity

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

\(\ds \paren {\lambda *_R \mu} \circ' x\) \(=\) \(\ds x \circ \paren {\lambda *_R \mu}\) Definition of $\circ’$
\(\ds \) \(=\) \(\ds x \circ \paren {\mu \times_R \lambda}\) Definition of $*_R$
\(\ds \) \(=\) \(\ds \paren {x \circ \mu} \circ \lambda\) Right Module Axiom $\text {RM} 3$: Associativity on $\struct {G, +_G, \circ}$
\(\ds \) \(=\) \(\ds \paren {\mu \circ' x} \circ \lambda\) Definition of $\circ’$
\(\ds \) \(=\) \(\ds \lambda \circ' \paren {\mu \circ' x}\) Definition of $\circ'$

$\blacksquare$


Also see


Sources

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