Editing Primitive ideal

{{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]].

== Primitive spectrum ==

The '''primitive spectrum''' of a ring is a non-commutative analog<ref group=note>A primitive ideal tends to be more of interest than a prime ideal in [[non-commutative ring theory]].</ref> of the [[prime spectrum]] of a commutative ring.

Let ''A'' be a ring and <math>\operatorname{Prim}(A)</math> the [[set (mathematics)|set]] of all primitive ideals of ''A''. Then there is a [[topological space|topology]] on <math>\operatorname{Prim}(A)</math>, called the '''[[Jacobson topology]]''', defined so that the [[closure (topology)|closure]] of a [[subset]] ''T'' is the set of primitive ideals of ''A'' containing the [[intersection (set theory)|intersection]] of elements of ''T''.

Now, suppose ''A'' is an [[associative algebra]] over a field. Then, by definition, a primitive ideal is the kernel of an [[irreducible representation]] <math>\pi</math> of ''A'' and thus there is a [[surjection]]
: <math>\pi \mapsto \ker \pi: \widehat{A} \to \operatorname{Prim}(A).</math>

Example: the [[spectrum of a C*-algebra|spectrum of a unital C*-algebra]].

== See also ==
* {{slink|Noncommutative algebraic geometry|History}}
* [[Dixmier mapping]]

== Notes ==
{{reflist|group=note}}

== References ==
* {{Citation | last1=Dixmier | first1=Jacques | title=Enveloping algebras | orig-date=1974 | url=https://books.google.com/books?isbn=0821805606 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=[[Graduate Studies in Mathematics]] | isbn=978-0-8218-0560-2 |mr=0498740 | year=1996 | volume=11}}
* {{Citation |last=Isaacs |first=I. Martin |year=1994 |title=Algebra |publisher=[[Brooks/Cole Publishing Company]] |isbn=0-534-19002-2}}

== External links ==
*{{cite web |title=The primitive spectrum of a unital ring |date=January 7, 2011 |work=[[Stack Exchange]] |url=https://math.stackexchange.com/q/16706 }}

[[Category:Ideals (ring theory)]]
[[Category:Module theory]]


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