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 analogA primitive ideal tends to be more of interest than a prime ideal in non-commutative ring theory. of the prime spectrum of a commutative ring.

Let ''A'' be a ring and $\operatorname(A)$ the set of all primitive ideals of ''A''. Then there is a topology on $\operatorname(A)$, 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 $\pi$ of ''A'' and thus there is a surjection
: $\pi \mapsto \ker \pi: \widehat \to \operatorname(A).$

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

See also

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* Dixmier mapping

Notes

References

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External links

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Ideals (ring theory)
Module theory

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