Definition:Trivial Module

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 $$\circ$$ be defined as:
 * $$\forall \lambda \in R: \forall x \in G: \lambda \circ x = e_G$$

Then $$\left({G, +_G, \circ}\right)_R$$ is an $R$-module.

Such a module is called a trivial module.

Unless $$R$$ is a ring with unity and $$G$$ contains only one element, this is not a unitary module.

Proof
Checking the criteria for module in turn:


 * $$(1) \quad \lambda \circ \left({x +_G y}\right) = e_G = e_G +_G e_G = \left({\lambda \circ x}\right) +_G \left({\lambda \circ y}\right)$$


 * $$(2) \quad \left({\lambda +_R \mu}\right) \circ x = e_G = e_G +_G e_G = \left({\lambda \circ x}\right) +_G \left({\mu \circ x}\right)$$


 * $$(3) \quad \left({\lambda \times_R \mu}\right) \circ x = e_G = \lambda \circ e_G = \lambda \circ \left({\mu \circ x}\right)$$

Thus the trivial module is indeed a module.

By definition, for the trivial module to be unitary, then $$R$$ 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 $$G$$ is the Trivial Group, and thus contains one element.