# Primitive ideal

In mathematics, a left primitive ideal in ring theory 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.

## Primitive spectrum

A primitive spectrum is a non-commutative analog[note 1] of a prime spectrum of a commutative ring.

Let A be a ring and $\operatorname {Prim} (A)$ the set of all primitive ideals of A. Then there is a topology on $\operatorname {Prim} (A)$ , 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 $\pi$ of A and thus there is a surjection

$\pi \mapsto \operatorname {ker} \pi :{\widehat {A}}\to \operatorname {Prim} (A).$ Example: a spectrum of a unital C*-algebra.