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December 26th, 2012, 12:17 PM  #1 
Senior Member Joined: Nov 2011 Posts: 100 Thanks: 0  If D is a division ring, why are Dmods semisimple?
Let D be a division ring. Why must every Dmodule be semisimple?

December 27th, 2012, 12:04 PM  #2 
Senior Member Joined: Aug 2010 Posts: 195 Thanks: 5  Re: If D is a division ring, why are Dmods semisimple?
If is a field, why must every module (ie vector space) be semisimple? What are the irreducible modules?

December 27th, 2012, 02:48 PM  #3 
Senior Member Joined: Mar 2012 Posts: 294 Thanks: 88  Re: If D is a division ring, why are Dmods semisimple?
if D is a division ring, then a Dmodule 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 Dmodule M as a direct sum: where: since D is a division ring, it has no nontrivial proper Dsubmodules (which would necessarily be nontrivial proper ideals of D). this is because every element of D{0} is a unit. 

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division, dmods, ring, semisimple 
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