Editing Simple module (section)

== The Jacobson density theorem ==
{{main|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 let ''D'' = End<sub>''R''</sub>(''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''&sdot;''A'' = ''x''&sdot;''r'' for all ''x'' in ''X''.<ref>Isaacs, Theorem 13.14, p. 185</ref>
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 ring|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]].