Intersection of Submodules is Submodule

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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 $M$.


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.