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<ref group=note>A primitive ideal tends to be more of interest than a prime ideal in non-commutative ring theory.</ref> of the prime spectrum of a commutative ring.

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

<math>\pi \mapsto \ker \pi: \widehat{A} \to \operatorname{Prim}(A).</math>

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

See also

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• Dixmier mapping

References

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External links

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