Intersection of Submodules is Submodule
Jump to navigation
Jump to search
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 $H$ and $K$ be submodules of $M$.
Then $H \cap K$ is also a submodule of $M$.
General Result
Let $S$ be a set of submodules of $M$.
Then the intersection $\ds \bigcap S$ is a submodule of $M$.
Proof
This is a special case of the General Result with $S = \set {H, K}$.
The proof follows immediately from the proof of the General Result.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 27$. Subspaces and Bases: Theorem $27.2$