Null Module Submodule of All

Theorem

Let $\struct {G, +_G, \circ}_R$ be an $R$-module.

Then the null module:

$\struct {\set {e_G}, +_G, \circ}_R$

is a submodule of $\struct {G, +_G, \circ}_R$.

Proof

From Trivial Subgroup is Subgroup, the trivial subgroup is a subgroup of the group $\struct {G, +_G}$.

The result follows from the Submodule Test.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 27$. Subspaces and Bases: Example $27.1$