Trivial Module is Not Unitary
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Theorem
Let $\struct {G, +_G}$ be an abelian group whose identity is $e_G$.
Let $\struct {R, +_R, \circ_R}$ be a ring.
Let $\struct {G, +_G, \circ}_R$ be the trivial $R$-module, such that:
- $\forall \lambda \in R: \forall x \in G: \lambda \circ x = e_G$
Then unless $R$ is a ring with unity and $G$ contains only one element, this is not a unitary module.
Proof
By definition, for a trivial module to be unitary, $R$ needs to be a ring with unity.
For Unitary Module Axiom $\text {UM} 4$: Unity of Scalar Ring to apply, we require that:
- $\forall x \in G: 1_R \circ x = x$
But for the trivial module:
- $\forall x \in G: 1_R \circ x = e_G$
So Unitary Module Axiom $\text {UM} 4$: Unity of Scalar Ring can apply only when:
- $\forall x \in G: x = e_G$
Thus for the trivial module to be unitary, it is necessary that $G$ be the trivial group, and thus to contain one element.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules: Example $26.6$