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