# Trivial Module is Not Unitary

## 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.

$\blacksquare$