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I'm looking for an article or source that deals with modules with this property:

$M$ is $R$-module which is direct sum of simple submodules all of which are isomorphic to simple module. I want to know more about this module and their properties.

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Such a module is called semisimple. –  Zhen Lin Aug 29 '12 at 11:49
    
I know this module is semisimple but I want to use this fact:all simple submodue is isomorphic. –  Babak Miraftab Aug 29 '12 at 12:07
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A bit of context would help. –  Julian Kuelshammer Aug 29 '12 at 12:37

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up vote 2 down vote accepted

Well in basic Wedderburn theory, given an $R$ module $M$, you call the sum of all simple submodules of $M$ which are isomorphic to a simple $R$ module $S$ a "homogeneous component of $M$".

For semisimple (Artinian) rings, the homogeneous components of the ring itself are exactly the simple rings in the Wedderburn decomposition.

I suppose one property you could draw from this is that their endomorphism rings are full linear rings over a division ring. (The division ring being the division ring of endomorphisms of $S$, which is common to all of the modules.)

And as others have noted they are obviously semisimple. What is nicer than semisimple? (Besides simple? and f.g. semisimple?)

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