Primitive ideal

Short description: Annihilator of a simple module

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.

## Contents

* 1 Primitive spectrum
* 2 See also
* 3 Notes
* 4 References
* 5 External links

## 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 Prim⁡(A) the set of all primitive ideals of A. Then there is a topology on 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 π of A and thus there is a surjection

π↦ker⁡π:A^->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, https://books.google.com/books?isbn=0821805606
* 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. https://math.stackexchange.com/q/16706.

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