Right Module over Ring Induces Left Module over Opposite Ring

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:

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

$(M \, 1)$ : Scalar Multiplication (Left) Distributes over Module Addition

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

 $\displaystyle \lambda \circ’ \paren{x +_G y}$ $=$ $\displaystyle \paren{x +_G y} \circ \lambda$ Definition of $\circ’$ $\displaystyle$ $=$ $\displaystyle x \circ \lambda +_G y \circ \lambda$ Right module axiom $(RM \, 1)$ on $\struct{G, +_G, \circ}$ $\displaystyle$ $=$ $\displaystyle \lambda \circ’ x +_G \lambda \circ’ y$ Definition of $\circ’$

$\Box$

$(M \, 2)$ : Scalar Multiplication (Right) Distributes over Scalar Addition

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

 $\displaystyle \paren {\lambda +_R \mu} \circ’ x$ $=$ $\displaystyle x \circ \paren {\lambda +_R \mu}$ Definition of $\circ’$ $\displaystyle$ $=$ $\displaystyle x \circ \lambda +_G x \circ \mu$ Right module axiom $(RM \, 2)$ on $\struct{G, +_G, \circ}$ $\displaystyle$ $=$ $\displaystyle \lambda \circ’ x +_G \mu \circ’ x$ Definition of $\circ’$

$\Box$

$(M \, 3)$ : Associativity of Scalar Multiplication

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

 $\displaystyle \paren {\lambda *_R \mu} \circ’ x$ $=$ $\displaystyle x \circ \paren {\lambda *_R \mu}$ Definition of $\circ’$ $\displaystyle$ $=$ $\displaystyle x \circ \paren {\mu \times_R \lambda}$ Definition of $*_R$ $\displaystyle$ $=$ $\displaystyle \paren {x \circ \mu} \circ \lambda$ Right module axiom $(RM \, 3)$ on $\struct{G, +_G, \circ}$ $\displaystyle$ $=$ $\displaystyle \paren {\mu \circ’ x} \circ \lambda$ Definition of $\circ’$ $\displaystyle$ $=$ $\displaystyle \lambda \circ’ \paren {\mu \circ’ x}$ Definition of $\circ’$

$\blacksquare$