Definition:Null Module

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

Let $$G$$ be the Trivial Group.

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

This module is known as the null module.

Proof
Follows from the fact that $$\left({G, +_G: \circ}\right)_R$$ has to be, by definition, a Trivial Module:

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