Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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
A bit of context would help. – Julian Kuelshammer Aug 29 '12 at 12:37
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?)

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.