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.
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.
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?)