Generated Submodule is Linear Combinations

Theorem
Let $$G$$ be a unitary $R$-module.

Let $$S \subseteq G$$.

Then the submodule $H$ generated by $S$ is the set of all linear combinations of $$S$$.

Proof

 * First the extreme case:

The smallest submodule of $$G$$ containing $$\varnothing$$ is $$\left\{{e_G}\right\}$$.

By definition, $$\left\{{e_G}\right\}$$ is the set of all linear combinations of $$\varnothing$$.


 * Now the general case:

Let $$\varnothing \subset S \subseteq G$$.

Let $$L$$ be the set of all linear combinations of $$S$$.

Since $$G$$ is a unitary $R$-module, every element $$x \in S$$ is the linear combination $$1_R x$$, so $$S \subseteq L$$.

But $$L$$ is closed for addition and scalar multiplication, so is a submodule.

Thus $$H \subseteq L$$.

But as every linear combination of $$S$$ clearly belongs to any submodule of $$G$$ which contains $$S$$, we also have $$L \subseteq H$$.