Left Module over Commutative Ring induces Right Module
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Theorem
Let $\struct {R, +_R, \times_R}$ be a commutative 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 $\struct {R, +_R, \times_R}$.
Proof
From Ring is Commutative iff Opposite Ring is Itself, $\struct {R, +_R, \times_R}$ is its own opposite ring.
From Left Module over Ring Induces Right Module over Opposite Ring, $\struct {G, +_G, \circ'}$ is a right module over $\struct {R, +_R, \times_R}$.
$\blacksquare$
Also see
Sources
- 2003: P.M. Cohn: Basic Algebra: Groups, Rings and Fields ... (previous) ... (next): Chapter $4$: Rings and Modules: $\S 4.1$: The Definitions Recalled