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December 26th, 2012, 12:17 PM   #1
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If D is a division ring, why are D-mods semisimple?

Let D be a division ring. Why must every D-module be semisimple?
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December 27th, 2012, 12:04 PM   #2
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Re: If D is a division ring, why are D-mods semisimple?

If is a field, why must every -module (ie vector space) be semisimple? What are the irreducible -modules?
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December 27th, 2012, 02:48 PM   #3
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Re: If D is a division ring, why are D-mods semisimple?

if D is a division ring, then a D-module is free (it satisfies, for example, all the usual axioms of a vector space). in particular, division rings have the invariant basis number property.

we can thus regard any D-module M as a direct sum:

where:



since D is a division ring, it has no non-trivial proper D-submodules (which would necessarily be non-trivial proper ideals of D). this is because every element of D-{0} is a unit.
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