Elements of Submodule form Subgroup
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Theorem
Let $\struct {R, +, \circ}$ be a ring.
Let $\struct {G, +_G}$ be an abelian group.
Let $\struct {G, +_G, \circ_G}_R$ be an $R$-module.
Let $\struct {H, +_H, \circ_H}_R$ be a submodule of $\struct {G, +_G, \circ_G}_R$.
Then $\struct {H, +_H}$ is a subgroup of $\struct {G, +_G}$.
Proof
By definition of submodule, $\struct {H, +_H}$ is an abelian group.
The result follows by definition of subgroup.
$\blacksquare$