Intersection and Sum of Submodules

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Theorem

Let $\left({G, +, \circ}\right)_R$ be an $R$-module.

Let $H$ and $K$ be submodules of $G$.


Then $H + K$ and $H \cap K$ are also submodules of $G$.

The intersection of any set of submodules of $G$ is a submodule.

Thus if $S \subseteq G$, the intersection of all submodules of $G$ containing $S$ is the smallest submodule of $G$ containing $S$.


Corollary

Let $\left({G, +, \circ}\right)_R$ be an $R$-module.

Ordered by $\subseteq$, the set of all submodules of $G$ is a complete lattice:

Let $H_1, H_2, \ldots, H_n$ be submodules of $G$.

Then:

$(1) \quad H_1 + H_2 + \cdots + H_n$ is the supremum
$(2) \quad H_1 \cap H_2 \cap \cdots \cap H_n$ is the infimum

of $\left\{{H_1, H_2, \ldots, H_n}\right\}$ in the complete lattice of all submodules of $G$.


Proof


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