Rings and sheaves
type: Item
instance of: scholarly article
title: Rings and sheaves
zbMATH Open document ID: 0880.46021
DOI: 10.1007/BF02362842
author: E. M. Vechtomov
published in: Journal of Mathematical Sciences (New York)
publication date: 1996-05-12T00:00:00Z
review text: This paper is a sequel to the author's earlier survey [Itogi Nauki Tekh., Ser. Algebra, Topologiya, Geom. 29, 119-191 (1991; Zbl 0835.54019)] whose goal is the representation of algebraic objects as sets of functions. Chapter 1 is a historical essay on some of the main theorems in rings of continuous functions, including those of Gelfand-Kolmogorov, Gelfand-Shilov, and Stone-Weierstrass. He also discusses ideals and subrings, including those of Gelfand and Wallman, as well as duality theorems and the Stone property. Chapter 2 is a historical essay and survey of sheaves of rings, leading to sheaves of rings of germs of continuous functions, as well as the spectra of rings. Chapter 3 treats the general theory of sheaf representations of rings, including those of Pierce and Lambek, among others. The survey ends with a bibliography of 550 items.
reviewed by: James V. Whittaker
Mathematics Subject Classification ID: 46E25
Mathematics Subject Classification ID: 54C40
Mathematics Subject Classification ID: 46-02
Mathematics Subject Classification ID: 46M20
Mathematics Subject Classification ID: 16S60
zbMATH DE Number: 877634
zbMATH Keywords: global sections of rings
zbMATH Keywords: representation of algebraic objects as sets of functions
zbMATH Keywords: rings of continuous functions
zbMATH Keywords: Gelfand-Kolmogorov, Gelfand-Shilov
zbMATH Keywords: Stone-Weierstrass
zbMATH Keywords: ideals
zbMATH Keywords: subrings
zbMATH Keywords: Stone property
zbMATH Keywords: sheaves of rings
zbMATH Keywords: germs of continuous functions
zbMATH Keywords: spectra of rings
MaRDI profile type: Publication
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cites work: Compact ringed spaces

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cites work: A generalisation of Gel'fand duality

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cites work: THE CONSTRUCTION OF SPACES DUAL TO PSEUDOCOMPLEMENTED DISTRIBUTIVE LATTICES

Rings and sheaves (cites work)
cites work: Q3222242

Rings and sheaves (cites work)
cites work: Q3923091

Rings and sheaves (cites work)
cites work: Concerning spaces of continuous functions

Rings and sheaves (cites work)
cites work: Q5554663

Rings and sheaves (cites work)
cites work: Sur la compacité du spectre minimal d'un anneau

Rings and sheaves (cites work)
cites work: Lindelöf realcompactifications

Rings and sheaves (cites work)
cites work: Alexandroff Algebras and Complete Regularity

Rings and sheaves (cites work)
cites work: Q3675163

Rings and sheaves (cites work)
cites work: Q3362560

Rings and sheaves (cites work)
cites work: Q4090509

Rings and sheaves (cites work)
cites work: Category theory applied to Pontryagin duality

Rings and sheaves (cites work)
cites work: Q4179411

Rings and sheaves (cites work)
cites work: Q5179949

Rings and sheaves (cites work)
cites work: On Structure Spaces of Ideals In Rings of Continuous Functions

Rings and sheaves (cites work)
cites work: A stucture sheaf for semiprime rings

Rings and sheaves (cites work)
cites work: The lambek sheaf reexamined

Rings and sheaves (cites work)
cites work: Spaces of vector-valued continuous functions

Rings and sheaves (cites work)
cites work: Compactness of the Structure Space of a Ring

Rings and sheaves (cites work)
cites work: Banach Bundles of Continuous Functions and an Integral Representation Theorem

Rings and sheaves (cites work)
cites work: What is a Sheaf?

Rings and sheaves (cites work)
cites work: Measures on F-Spaces

Rings and sheaves (cites work)
cites work: Q4092505

Rings and sheaves (cites work)
cites work: Non-Archimedean Gelfand theory

Rings and sheaves (cites work)
cites work: Non-Archimedean Banach algebras

Rings and sheaves (cites work)
cites work: Prime Ideals and Sheaf Representation of a Pseudo Symmetric Ring

Rings and sheaves (cites work)
cites work: On semigroups of continuous transformations of fuzzily structured sets

Rings and sheaves (cites work)
cites work: Q3290875

Rings and sheaves (cites work)
cites work: Reticulated rings

Rings and sheaves (cites work)
cites work: Torsion theoretic points and spaces

Rings and sheaves (cites work)
cites work: Q3697187

Rings and sheaves (cites work)
cites work: Sheaf representations of strongly harmonic rings

Rings and sheaves (cites work)
cites work: Compact representations - the lattice theory of compact ringed spaces

Rings and sheaves (cites work)
cites work: On rings of sets

Rings and sheaves (cites work)
cites work: On rings of sets II. Zero-sets

Rings and sheaves (cites work)
cites work: Spaces of ideals of distributive lattices II. Minimal prime ideals

Rings and sheaves (cites work)
cites work: A sheaf theoretic representation of rings with Boolean orthogonalities

Rings and sheaves (cites work)
cites work: The Theory of Representation for Boolean Algebras

Rings and sheaves (cites work)
cites work: Applications of the Theory of Boolean Rings to General Topology

Rings and sheaves (cites work)
cites work: Topological representations of distributive lattices and Brouwerian logics

Rings and sheaves (cites work)
cites work: A topological structure on the structure shaf of a topological ring

Rings and sheaves (cites work)
cites work: Wallman-Type Compactifications on 0-Dimensional Spaces

Rings and sheaves (cites work)
cites work: Q5572410

Rings and sheaves (cites work)
cites work: Noncommutative rings in which every prime ideal is contained in a unique maximal ideal

Rings and sheaves (cites work)
cites work: Triple and sheaf representations of Stone lattices

Rings and sheaves (cites work)
cites work: Vector Bundles and Projective Modules

Rings and sheaves (cites work)
cites work: Algebraic \(K\)-theory

Rings and sheaves (cites work)
cites work: Q4106354

Rings and sheaves (cites work)
cites work: Q4163353

Rings and sheaves (cites work)
cites work: On an exact sequence of modules

Rings and sheaves (cites work)
cites work: The sheaf representation of near-rings and its applications

Rings and sheaves (cites work)
cites work: On Sheaf Representation of a Biregular Near-Ring

Rings and sheaves (cites work)
cites work: Q3892392

Rings and sheaves (cites work)
cites work: Q4156532

Rings and sheaves (cites work)
cites work: Q5562063

Rings and sheaves (cites work)
cites work: Q4106357

Rings and sheaves (cites work)
cites work: Q4098225

Rings and sheaves (cites work)
cites work: Q4107501

Rings and sheaves (cites work)
cites work: Space of minimal prime ideals of a ring without nilpotent elements

Rings and sheaves (cites work)
cites work: Prime strict ideals in Rickart *-rings

Rings and sheaves (cites work)
cites work: Arens Regularity of the Algebra C(K,A )

Rings and sheaves (cites work)
cites work: Q4085851

Rings and sheaves (cites work)
cites work: The affine scheme associated to a non-Noetherian prime PI algebra

Rings and sheaves (cites work)
cites work: Q3778858

Rings and sheaves (cites work)
cites work: Local cohomology and quasicoherent sheaves

Rings and sheaves (cites work)
cites work: Q3800224

Rings and sheaves (cites work)
cites work: Q4135749

Rings and sheaves (cites work)
cites work: Q5682778

Rings and sheaves (cites work)
cites work: Characters of Cartesian Products of Algebras

Rings and sheaves (cites work)
cites work: Dimension functions on the prime spectrum

Rings and sheaves (cites work)
cites work: The maximal ideal space of a Noetherian ring

Rings and sheaves (cites work)
cites work: Intersections of Primary Ideals in Rings of Continuous Functions

Rings and sheaves (cites work)
cites work: Q5181410

Rings and sheaves (cites work)
cites work: Banach Algebras of Continuous Functions

Rings and sheaves (cites work)
cites work: Structure spaces of rings and Banach algebras

Rings and sheaves (cites work)
cites work: Projective modules over rings of global sections of flasque sheaves

Rings and sheaves (cites work)
cites work: Some remarks on p-rings and their Boolean geometry