Editing Primitive ideal (section)

{{Short description|Annihilator of a simple module}}
{{Distinguish|primary ideal|principal ideal}}

In [[mathematics]], specifically [[ring theory]], a left '''primitive ideal''' is the [[Annihilator (ring theory)|annihilator]] of a (nonzero) [[simple module|simple]] left [[module (mathematics)|module]]. A right primitive ideal is defined similarly. Left and right primitive ideals are always two-sided ideals.

Primitive ideals are [[prime ideal|prime]]. The [[quotient ring|quotient]] of a [[ring (mathematics)|ring]] by a left primitive ideal is a left [[primitive ring]]. For [[commutative ring]]s the primitive ideals are [[maximal ideal|maximal]], and so commutative primitive rings are all [[field (mathematics)|fields]].