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primitive ideal

{{Short description|Annihilator of a simple module}}

{{Distinguish|primary ideal|principal 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{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.

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{A} \to \operatorname{Prim}(A).

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

See also

Notes

{{reflist|group=note}}

References

External links

  • {{cite web |title=The primitive spectrum of a unital ring |date=January 7, 2011 |work=Stack Exchange |url=https://math.stackexchange.com/q/16706 }}

Category:Ideals (ring theory)

Category:Module theory

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