The prime spectrum of a ring (associative, but not necessarily commutative or unital) is the set of prime ideals of the ring; thus it generalizes the (underlying topological space of) spectrum of a commutative ring. For noncommutative rings s (and noncommutative C-star-algebras) however sometimes spectra of primitive ideal?s are more interesting. This entry should eventually be more about the noncommutative case.
See more at spectrum of a commutative ring.
The prime spectrum of a commutative unital ring extends to a contravariant functor .
The prime spectrum of a commutative unital ring has a natural Zariski topology , given by the basis of open sets where . The prime spectrum considered as a topological space is also called Zariski spectrum. A ring map induces a continuous map , so there is a contravariant functor .
The correspondence for all extends to a unique sheaf of commutative local rings on the Zariski topology on . The ringed space so constructed is also called the prime spectrum of the commutative ring . An affine scheme as a locally ringed space is any ringed space which is isomorphic to the prime spectrum of a commutative ring.
Any morphism of commutative rings also induces the comorphism of structure sheaves on spectra, hence a morphism in locally ringed spaces. This way one obtains a contravariant functor which is fully faithful (and its essential image is the strictly full subcategory whose objects are affine schemes).
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S. Launois, T. H. Lenagan, L. Rigal, Prime ideals in the quantum grassmannian, Sel. math., New ser. 13, 697 (2008) doi
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Last revised on July 23, 2024 at 14:24:35. See the history of this page for a list of all contributions to it.