Intersection and Sum of Submodules
into Intersection of Submodules is Submodule and Sum of Submodules is Submodule
Until this has been finished, please leave {{refactor}} in the code.
New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.
Contents
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
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues.
To discuss this proof in more detail, feel free to use the talk page.
If you are able to fix this page, then when you have done so you can remove this instance of {{Questionable}} from the code.
If you would welcome a second opinion as to whether your correction is correct, add a call to {{Proofread}} the page (see the proofread template for usage).
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
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
When this page/section has been completed, {{ProofWanted}} should be removed from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page (see the proofread template for usage).
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): $\S 27$: Theorem $27.2$
