Simple Module - The Jacobson Density Theorem

The Jacobson Density Theorem

An important advance in the theory of simple modules was the Jacobson density theorem. The Jacobson density theorem states:

Let U be a simple right R-module and write D = End_R(U). Let A be any D-linear operator on U and let X be a finite D-linearly independent subset of U. Then there exists an element r of R such that x·A = x·r for all x in X.

In particular, any primitive ring may be viewed as (that is, isomorphic to) a ring of D-linear operators on some D-space.

A consequence of the Jacobson density theorem is Wedderburn's theorem; namely that any right artinian simple ring is isomorphic to a full matrix ring of n by n matrices over a division ring for some n. This can also be established as a corollary of the Artin–Wedderburn theorem.