Intersection of Set of Submodules containing Subset is Smallest Submodule

Theorem

Let $R$ be a ring.

Let $\struct {G, +_G}$ be an abelian group.

Let $M = \struct {G, +, \circ}_R$ be an $R$-module.

Let $S \subset M$ be a subset of $M$.

Let $T$ be the set of all submodules of $M$ which contain $S$ as a subset.

Then the intersection $\bigcap T$ is the smallest submodule of $M$ containing $S$.

Proof

By hypothesis, we have:

$\ds T = \bigcap \set { M' \subseteq M : S \subseteq M', \textrm {$M'$ is a submodule of $M$} }$

From Intersection of Submodules is Submodule:General Result, it follows that $\bigcap T$ is a submodule of $M$.

As $S \subseteq M'$ for all $M' \in T$, it follows that $S \subseteq \bigcap T$.

Let $M' \in T$ be a submodule of $M$ such that $S \subseteq M'$.

From Intersection is Subset:General Result, it follows that $M' \subseteq \bigcap T$.

Hence the result.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 27$. Subspaces and Bases: Theorem $27.2$