Null Module Submodule of All
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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 $27.1$