Definition:Trivial Module

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

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

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

Then $$\left({G, +: \circ}\right)_R$$ 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.