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

From ProofWiki
Jump to navigation Jump to search

Theorem

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

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 $\struct {R, +_R, \times_R}$ if and only if:

$\forall \lambda, \mu \in R: \forall x \in G: x \circ \paren{ \lambda \times_R \mu} = x \circ \paren{\mu \times_R \lambda}$

Proof

Necessary Condition

Let $\struct{G, +_G, \circ’}$ be a left module over $\struct {R, +_R, \times_R}$.

Then:

\(\displaystyle x \circ \paren{ \lambda \times_R \mu}\) \(=\) \(\displaystyle \paren{ \lambda \times_R \mu} \circ’ x\) Definition of $\circ’$
\(\displaystyle \) \(=\) \(\displaystyle \lambda \circ’ \paren{\mu \circ’ x}\) Left module axiom $(M \, 3)$ : Associativity of Scalar Multiplication
\(\displaystyle \) \(=\) \(\displaystyle \paren {x \circ \mu} \circ \lambda\) Definition of $\circ’$
\(\displaystyle \) \(=\) \(\displaystyle x \circ \paren {\mu \times_R \lambda}\) Right module axiom $(RM \, 3)$ : Associativity of Scalar Multiplication

$\Box$

Sufficient Condition

Let the scalar multiplication $\circ$ satisfy:

$\forall \lambda, \mu \in R: \forall x \in G: x \circ \paren{ \lambda \times_R \mu} = x \circ \paren{\mu \times_R \lambda}$

It needs to be shown that $\struct{G, +_G, \circ’}$ satisfies the left module axioms.


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

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

Then:

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

$\Box$

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

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

Then:

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

$\Box$

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

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

Then:

\(\displaystyle \paren{\lambda \times_R \mu} \circ’ x\) \(=\) \(\displaystyle x \circ \paren{\lambda \times_R \mu}\) Definition of $\circ’$
\(\displaystyle \) \(=\) \(\displaystyle x \circ \paren{\mu \times_R \lambda}\) Assumption
\(\displaystyle \) \(=\) \(\displaystyle \paren{x \circ \mu} \circ \lambda\) Right module axiom $(RM \, 3)$ on $\circ$
\(\displaystyle \) \(=\) \(\displaystyle \lambda \circ’ \paren {\mu \circ’ x}\) Definition of $\circ’$

$\blacksquare$

Also see