Editing Primitive ring (section)

== Definition ==
A ring ''R'' is said to be a '''left primitive ring''' if it has a [[faithful module|faithful]] [[simple module|simple]] left [[module (mathematics)|''R''-module]].  A '''right primitive ring''' is defined similarly with right ''R''-modules. There are rings which are primitive on one side but not on the other. The first example was constructed by [[George Bergman|George M. Bergman]] in {{harv|Bergman|1964}}.  Another example found by Jategaonkar showing the distinction can be found in {{harvtxt|Rowen|1988|p=159}}.

An internal characterization of left primitive rings is as follows: a ring is left primitive if and only if there is a [[maximal ideal|maximal left ideal]] containing no nonzero two-sided [[ideal (ring theory)|ideals]]. The analogous definition for right primitive rings is also valid.

The structure of left primitive rings is completely determined by the [[Jacobson density theorem]]: A ring is left primitive if and only if it is [[ring homomorphism|isomorphic]] to a [[Jacobson density theorem#Topological characterization|dense]] [[subring]] of the [[ring of endomorphisms]] of a [[Division_ring#Relation_to_fields_and_linear_algebra|left vector space]] over a [[division ring]].

Another equivalent definition states that a ring is left primitive if and only if it is a [[prime ring]] with a faithful left module of [[length of a module|finite length]] ({{harvnb|Lam|2001}}, [https://books.google.com/books?id=2T5DAAAAQBAJ&pg=PA191 Ex.&nbsp;11.19, p.&nbsp;191]).