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 $\O$ is $\set {e_G}$.

By definition of linear combination of empty set, $\set {e_G}$ is the set of all linear combinations of $\O$.

Now the general case:

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

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 $L \subseteq H$.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 27$. Subspaces and Bases: Theorem $27.3$