Understanding the modules of semiprimitive rings 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.
 A: You write

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

but in general the quotient of a ring by a left ideal isn't a ring
(it is when the ideal is two-sided but that won't usually be the
case when the ring is noncommutative).
In the commutative case, then yes, a semiprimitive $R$ does embed into
$S=\prod_j R_j=\prod_j R/I_j$ where $I_j$ runs over the maximal ideals of $R$.
But alas, I cannot see a concrete question here. As a simple example, consider
$R=\mathbb{Z}$. Then $S=\prod_p(\mathbb{Z}/p\mathbb{Z})$ but I cannot
see how looking at $S$ helps with understanding the $\mathbb{Z}$-modules
$\mathbb{Z}$ or $\mathbb{Q}$ or the plethora of modules in between these two.
A: No! The Weyl algebra is simple. The problem of classifying the finite dimensional representations is wild. This was discussed in
Is there a machinery describing all the irreducible representations ?
In case this is too succinct. The Weyl algebra is the algebra of linear differential operators with polynomial coefficients. It is generated by $x$ and $D$ with defining relation $Dx-xD=1$. This acts on polynomials in $x$ with $D=d/dx$. This is a faithful simple representation.
The problem of finding finite dimensional representations is the problem of solving linear differential equations.
