Trivial Module is Not Unitary

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Theorem

Let $\left({G, +_G}\right)$ be an abelian group whose identity is $e_G$.

Let $\left({R, +_R, \circ_R}\right)$ be a ring.


Let $\left({G, +_G, \circ}\right)_R$ 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 Module: $(4)$ 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 Module: $(4)$ 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.

$\blacksquare$


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