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.

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.

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.)

Here's a bit of background. Perhaps my question is badly formed, in which case perhaps someone can tell me where I've got confused.

A ring is semiprimitive when it has zero Jacobson radical, or equivalently when it has a faithful semisimple module. Given a semiprimitive ring, I'm pretty sure that this faithful semisimple module can be taken to be the direct product of all of the simple modules. I'm also pretty sure that these simple modules can be obtained by taking a quotient of the ring by maximal left ideals, and so these simple modules are themselves also rings. So, writing $R$ for our semiprimitive ring and $R_i$ for the rings which are its simple modules, we have an injective ring homomorphism $$f: R \to \Pi_i R_i$$ which is surjective on each of the factors. This construction is called the subdirect product. Can we somehow understand ALL of the modules for $R$ in terms of $\Pi_i R_i$?

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.