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-   -   If D is a division ring, why are D-mods semisimple? (http://mymathforum.com/abstract-algebra/32831-if-d-division-ring-why-d-mods-semisimple.html)

watson December 26th, 2012 12:17 PM

If D is a division ring, why are D-mods semisimple?
 
Let D be a division ring. Why must every D-module be semisimple?

Turgul December 27th, 2012 12:04 PM

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?

Deveno December 27th, 2012 02:48 PM

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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