The Artin-Wedderburn theorem shows that the definition of a semisimple ring is enormously restrictive. Even fails to be semisimple! A less restrictive notion, but one that still captures the notion of a ring which can be understood by how it acts on simple (left) modules, is that of a semiprimitive or Jacobson semisimple ring, one with the property that every element acts nontrivially in some simple (left) module .
Said another way, let the Jacobson radical of a ring consist of all elements of which act trivially on every simple module. By definition, this is an intersection of kernels of ring homomorphisms, hence a two-sided ideal. A ring is then semiprimitive if it has trivial Jacobson radical.
The goal of this post will be to discuss some basic properties of the Jacobson radical. I am again working mostly from Lam’s A first course in noncommutative rings.