Algebra in a Localic Topos with Applications to Ring Theory

Borceux, Francis; Bossche, Gilberte Van den

doi:10.1007/bfb0073030

Cited by 42 publications

(26 citation statements)

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“…Thus, we can represent R as the global elements of a sheaf X over the idempotent spectrum EIdl(R). Restricting X to the neat ideals of R we obtain the pure representation of R studied by Borceux and Van Den Bossche [2] and hence restricting X further to the ideals generated by idempotents we obtain the Pierce representation.…”

Section: Resultsmentioning

confidence: 99%

“…In this paper, we shall show that the same is true for the Pierce representation and also for the pure representation of Borceux and Van Den Bossche [2], provided that the equality data is given in the quantale Idl(R) of ideals of R.…”

Section: Introductionmentioning

confidence: 94%

“…We recall from [11] that the neat ideals form a localic subquantale of Idl(R) which we denote NIdl(R). The ring has a sheaf representation R over this locale given by taking/~(E) to be the set of R-module homomorphisms h: E -+ R [2] (see [7, pp. 188-191] for details).…”

Section: Related Representationsmentioning

confidence: 99%

“…Borceux et al have already written several papers outlining a theory of sheaves on various sorts of quantale [1,2]. The most recent [4] sketches the way in which these sheaves might provide a generic representation theorem for rings (see also [9]).…”

Section: Introductionmentioning

confidence: 99%

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Generalized logic and the representation of rings

Ambler

Verity

1996

Appl Categor Struct

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Following the progression towards weaker logics, a number of authors have considered the notion of a 'sheaf over a quantale' or, equivalently, a 'quantale valued set'. In this paper, we use ideas from enriched category theory to motivate the definition of a 'quantic sheaf'. Given a localic subquantale of Q, a quantic sheaf over Q gives a sheaf in the usual sense. As an application, we derive a series of sheaf representations for commutative rings including the familiar Pierce representation.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 94%

Section: Related Representationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Generalized logic and the representation of rings

Ambler

Verity

1996

Appl Categor Struct

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“…A ring R is called Gelfand (see Borceux and Van den Bossche, 1983) if, for each pair of distinct maximal right ideals M 1 , M 2 of R, there are right ideals I 1 , I 2 of R, such that I 1 M 1 , I 2 M 2 and I 1 I 2 = 0. In this article, we shall prove that a von Neumann regular ring R is a Gelfand ring if and only if it is an Abelian ring if and only if it is a right or left top ring if and only if it is a right or left pm-ring (Theorem 2.12).…”

Section: Introductionmentioning

confidence: 99%

Spectrum of a Noncommutative Ring

Zhang

Tong

Wang

2006

Communications in Algebra

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R is any ring with identity. Let Spec r R resp. Spec R be the set of all prime right ideals resp. all prime ideals of R and let U r eR = P ∈ Spec r R e P . In this article, we study the relationships among various ring-theoretic properties and topological conditions on Spec r R with weak Zariski topology . A ring R is called Abelian if all idempotents in R are central see Goodearl, 1991 . A ring R is called 2-primal if every nilpotent element is in the prime radical of R see Lam, 2001 . It will be shown that for an Abelian ring R there is a bijection between the set of all idempotents in R and the clopen i.e., closed and open sets in Spec r R . And the following results are obtained for any ring R: 1 For any clopen set U in Spec r R , there is an idempotent e in R such that U = U r eR . 2 If R is an Abelian ring or a 2-primal ring, then, for any idempotent e in R, U r eR is a clopen set in Spec r R . 3 Spec r R is connected if and only if Spec R is connected.

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