Editing Primitive ideal (section)

== 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]].