As far as I understand, a semiprimitive ring can be fully 'explored' by its simple modules, in the sense that a semiprimitive ring is the subdirect product of its simple modules (for brevity, I'll use 'module' to mean 'left module' throughout.) This is great, and it clearly tells us a lot about the ring. However, we can only claim to fully understand a ring when we know ALL of its modules, not just its simple ones. So can we use this subdirect product decomposition to help us characterize all of the modules for the ring, beyond the ones which are products of simple modules? (Note that a semiprimitive ring is not necessarily semisimple.)

EDIT: It seems I made a mistake in the original post, it's not true that a semiprimitive ring is a subdirect product of its simple modules. But it IS always the semidirect product of SOME list of primitive rings. (Is a minimal such list of primitive rings uniquely defined?) So, the question is to what extent understanding the representation theory of these primitive rings helps you with understanding the representation theory of the original ring.