# Null Module is Module

## Theorem

Let $\struct {R, +_R, \circ_R}$ be a ring.

Let $G$ be the trivial group.

Let $\struct {G, +_G, \circ}_R$ be the null module.

Then $\struct {G, +_G, \circ}_R$ is a module.

## Proof

Follows from the fact that $\struct {G, +_G, \circ}_R$ has to be, by definition, a trivial module:

$\circ$ can only be defined as:

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

$\blacksquare$