Left Module over Ring Induces Right 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 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 $\struct {R, +_R, *_R}$.


Proof

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

By definition of the opposite ring:

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


Right Module Axiom $\text {RM} 1$: (Right) Distributivity over Module Addition

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

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

$\Box$


Right Module Axiom $\text {RM} 2$: (Left) Distributivity over Scalar Addition

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

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

$\Box$


Right Module Axiom $\text {RM} 3$: Associativity

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

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

$\blacksquare$


Also see


Sources

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