Semiprime Rings

As was the case with groups, a ring is said to be simple if it has no proper homomorphic images (equivalently, it has no proper 2-sided ideals). On the other hand, a right (or left) R-module without proper homomorphic images is said to be irreducible. A right (or left) module is said to be completely reducible is it is a direct sum of irreducible modules. Similarly, a ring R is said to be completely reducible if and only if the right module $$R_R$$ R R is completely reducible. A ring is semiprimitive if and only if the intersection of its maximal ideals is zero, and is semiprime if and only if the intersection of all its prime ideals is the zero ideal. Written in the presented order, each of these three properties of rings implies its successor—that is, the properties become weaker. The goal here is to prove the Artin-Wedderburn theorem, basically the following two statements: (1) A ring is completely reducible if and only if it is a direct sum of finitely many full matrix algebras, each summand defined over its own division ring. (2) If R is semiprimitive and Artinian (i.e. it has the descending chain condition on right ideals) then the same conclusion holds. A corollary is that any completely reducible simple ring is a full matrix algebra.