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modular law (Theorem)

Let $ _RM$ be a left $ R$-module with submodules $ A, B, C$, and suppose $ C \subseteq B$. Then

$\displaystyle C + (B \cap A) = B \cap (C+A) $

This result shows that the submodules of $ _RM$, partially ordered by inclusion, form a modular lattice with $ \cap$ as the meet and $ +$ as the join.



"modular law" is owned by yark. [ full author list (2) | owner history (1) ]
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See Also: modular lattice


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proof of modular law (Proof) by yark
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Cross-references: meet, modular lattice, inclusion, submodules
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This is version 4 of modular law, born on 2002-07-18, modified 2006-12-20.
Object id is 3173, canonical name is ModularLaw.
Accessed 2931 times total.

Classification:
AMS MSC: 16D10 (Associative rings and algebras :: Modules, bimodules and ideals :: General module theory)
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