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.
Primitive spectrum.
The primitive spectrum of a ring is a non-commutative analog of the prime spectrum of a commutative ring.
Let "A" be a ring and formula_1 the set of all primitive ideals of "A". Then there is a topology on formula_1, 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 formula_3 of "A" and thus there is a surjection
Example: the spectrum of a unital C*-algebra.