Primitive ideal explained

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^[1] of the 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.\pi of A and thus there is a surjection\pi \mapsto \ker \pi: \widehat{A} \to \operatorname{Prim}(A).

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

See also

• • Dixmier mapping

External links

• Web site: The primitive spectrum of a unital ring . January 7, 2011 . .

Notes and References

1. A primitive ideal tends to be more of interest than a prime ideal in non-commutative ring theory.