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.

Not to be confused with primary ideal or principal ideal.

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[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

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

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

See also

• noncommutative algebraic geometry#History
• Dixmier mapping

Notes

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

References

• Dixmier, Jacques (1996) [1974], Enveloping algebras, Graduate Studies in Mathematics, 11, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0560-2, MR 0498740
• Isaacs, I. Martin (1994), Algebra, Brooks/Cole Publishing Company, ISBN 0-534-19002-2

External links

• "The primitive spectrum of a unital ring". Stack Exchange. January 7, 2011.