An Artinian ring is initially understood via its largest semisimple quotient.
A module over a (not necessarily commutative) ring is said to be semisimple (or completely reducible) if it is the direct sum of simple (irreducible) submodules.
On the other hand, the ring Z of integers is not a semisimple module over itself, since the submodule 2Z is not a direct summand.
Semisimple is stronger than completely decomposable, which is a direct sum of indecomposable submodules.
From this it follows that or in more exact terms In particular, any module over a semisimple ring is injective and projective.
These and many other nice examples are discussed in more detail in several noncommutative ring theory texts, including chapter 3 of Lam's text, in which they are described as nonartinian simple rings.
The module theory for the Weyl algebras is well studied and differs significantly from that of semisimple rings.