Elements of Submodule form Subgroup

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.