Left Module induces Right Module over same Ring iff Actions are Commutative
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Contents
Theorem
Let $\struct {R, +_R, \times_R}$ be a 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 $\struct {R, +_R, \times_R}$ if and only if:
- $\forall \lambda, \mu \in R: \forall x \in G: \paren{ \lambda \times_R \mu} \circ x = \paren{\mu \times_R \lambda} \circ x$
Proof
Necessary Condition
Let $\struct{G, +_G, \circ’}$ be a right module over $\struct {R, +_R, \times_R}$.
Then:
| \(\displaystyle \paren{ \lambda \times_R \mu} \circ x\) | \(=\) | \(\displaystyle x \circ’ \paren{ \lambda \times_R \mu}\) | Definition of $\circ’$ | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \paren {x \circ’ \lambda} \circ’ \mu\) | Right module axiom $(RM \, 3)$ : Associativity of Scalar Multiplication | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \mu \circ \paren {\lambda \circ x}\) | Definition of $\circ’$ | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \paren {\mu \times_R \lambda} \circ x\) | Left module axiom $(M \, 3)$ : Associativity of Scalar Multiplication |
$\Box$
Sufficient Condition
Let the scalar multiplication $\circ$ satisfy:
- $\forall \lambda, \mu \in R: \forall x \in G: \paren{ \lambda \times_R \mu} \circ x = \paren{\mu \times_R \lambda} \circ x$
It needs to be shown that $\struct{G, +_G, \circ’}$ satisfies the right module axioms.
$(RM \, 1)$ : Scalar Multiplication (Right) Distributes over Module Addition
Let $\lambda, \mu \in R, x \in G$.
Then:
| \(\displaystyle \paren {x +_G y} \circ’ \lambda\) | \(=\) | \(\displaystyle \lambda \circ \paren {x +_G y}\) | Definition of $\circ’$ | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \lambda \circ x +_G \lambda \circ y\) | Left module axiom $(M \, 1)$ on $\circ$ | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle x \circ’ \lambda +_G y \circ’ \lambda\) | Definition of $\circ’$ |
$\Box$
$(RM \, 2)$ : Scalar Multiplication (Left) Distributes over Scalar Addition
Let $\lambda \in R, x, y \in G$.
Then:
| \(\displaystyle x \circ’ \paren{\lambda +_R \mu}\) | \(=\) | \(\displaystyle \paren{\lambda +_R \mu} \circ x\) | Definition of $\circ’$ | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \lambda \circ x +_G \mu \circ y\) | Left module axiom $(M \, 2)$ on $\circ$ | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle x \circ’ \lambda +_G x \circ’ \mu\) | Definition of $\circ’$ |
$\Box$
$(RM \, 3)$ : Associativity of Scalar Multiplication
Let $\lambda, \mu \in R, x \in G$.
Then:
| \(\displaystyle x \circ’ \paren{\lambda \times_R \mu}\) | \(=\) | \(\displaystyle \paren{\lambda \times_R \mu} \circ x\) | Definition of $\circ’$ | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \paren{\mu \times_R \lambda} \circ x\) | Assumption | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \mu \circ \paren{\lambda \circ x}\) | Left module axiom $(M \, 3)$ on $\circ$ | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \paren {x \circ’ \lambda} \circ’ \mu\) | Definition of $\circ’$ |
$\blacksquare$
