Primitive ideal

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

Primitive ideals are prime. The quotient of a ring by a left primitive ideal is a left primitive ring. For commutative rings the primitive ideals are maximal, and so commutative primitive rings are all fields.

The primitive spectrum of a ring is a non-commutative analog^{note 1} of the prime spectrum of a commutative ring.

Let A be a ring and ${\displaystyle \operatorname {Prim} (A)}$ the set of all primitive ideals of A. Then there is a topology on ${\displaystyle \operatorname {Prim} (A)}$, called the Jacobson topology, defined so that the closure of a subset T is the set of primitive ideals of A containing the 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 ${\displaystyle \pi }$ of A and thus there is a surjection

${\displaystyle \pi \mapsto \ker \pi :{\widehat {A}}\to \operatorname {Prim} (A).}$

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