Intersection and Sum of Submodules
into Intersection of Submodules is Submodule and Sum of Submodules is Submodule
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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:
a complete lattice is not characterised by using finitely many $H$ only; one needs to proceed to infinite sets, using the standard vector space-like approach of span (finite combination of terms from the $H_i$) for the addition, I suspect
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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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Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): $\S 27$: Theorem $27.2$