Equivalence of Definitions of Generator of Unitary Module

Theorem
Let $R$ be a ring with unity.

Let $M$ be a unitary $R$-module.

Let $S \subseteq M$ be a subset.

Definition by linear combinations
$S$ is a generator of $M$ every element of $M$ is a linear combination of elements of $S$.

Definition by intersection of submodules
That is, $M$ is equal to the intersection of all submodules of $M$ containing $S$.

Definition by intersection of submodules implies Definition by linear combinations
By definition of submodule, it follows that $M$ can be considered as a submodule of itself.

From Equivalence of Definitions of Generated Submodule over Ring with Unity, it follows that:


 * the set of all linear combinations of elements of $S$

is equal to:


 * the intersection of all submodules of $M$ containing $S$.

Hence, the two definitions of generator of unitary module by linear combinations and by intersection of submodules are equivalent.

Definition by proper submodules implies Definition by intersection of submodules
By definition of module, it follows that $M$ is an $R$-module.

The equivalence of the definitions now follows from Equivalence of Definitions of Generator of Module.