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.

The following definitions of the concept of Generator of Unitary Module are equivalent:

Definition by linear combinations

$S$ is a generator of $M$ if and only if every element of $M$ is a linear combination of elements of $S$.

Definition by intersection of submodules

$S$ is a generator of $M$ if and only if $M$ is the submodule generated by $S$.

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

Definition by proper submodules

$S$ is a generator of $M$ if and only if $M$ has no proper submodule containing $S$.

Proof

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.

$\Box$

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.

$\blacksquare$