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Topic: Category of commutative rings

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Spectrum of a ring

Zariski topology

Julia Craven Howard

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	Learn more about Category theory in the online encyclopedia. (Site not responding. Last check: 2007-11-06)
		Algebra of continuous functions: a contravariant functor from the category of topological spaces (with continuous maps as morphisms) to the category of real associative algebras is given by assigning to every topological space X the algebra C(X) of all real-valued continuous functions on that space.
		One of the central themes of algebraic geometry is the equivalence of the category C of affine schemes and the category D of commutative rings.
		Another important duality occurs in functional analysis: the category of commutative C-algebras with identity is contravariantly equivalent to the category* of compact Hausdorff spaces.
	www.onlineencyclopedia.org /c/ca/category_theory.html (2963 words)

	Rings
		The theory of commutative rings resembles the theory of numbers in several respects, and various definitions for commutative rings are designed to recover properties known from the integers.
		In commutative ring theory, numbers are often replaced by ideals, and the definition of prime ideal tries to capture the essence of prime numbers.
		A module over a ring is an abelian group that the ring acts on as a ring of endomorphisms, very much akin to the way fields (integral domains in which every non-zero element is invertible) act on vector spaces.
	www.risberg.ws /Hypertextbooks/Mathematics/Algebra/rings.htm (890 words)

	Business Encyclopedia (Site not responding. Last check: 2007-11-06)
		Suppose X is a topological space, and C is a category (often, this is the category of sets, the category of Abelian groups, the category of commutative rings, or the category of modules over a fixed ring).
		Ringed spaces are sheaves of commutative rings; especially important are the locally ringed spaces where all stalks (see below) are local rings.
		is an object of C, for C a category such as the category of abelian groups or the category of commutative rings.
	www.bizencyclopedia.com /index.php?title=Sheaf (2898 words)

	Kids.net.au - Encyclopedia Category theory -
		Category theory is half-jokingly known as "abstract nonsense".
		Category theory is also used in a foundational way in functional programming, for example to discuss the idea of typed lambda calculus in terms of cartesian-closed categories.
		One may check that the map from the category of Hausdorff topological spaces with a distinguished point to the category of groups is functorial: a topological (homo/iso)morphism will naturally correspond to a group (homo/iso)morphism.
	www.kids.net.au /encyclopedia-wiki/ca/Category_theory (2107 words)

	13: Commutative rings and algebras (Site not responding. Last check: 2007-11-06)
		Of particular interest are several classes of rings of interest in number theory, field theory, algebraic geometry, and related areas; however, other classes of rings arise, and a rich structure theory arises to analyze commutative rings in general, using the concepts of ideals, localizations, and homological algebra.
		Conversely, the study of a ring is often focused by the examination of related fields, such as the quotients by each of the maximal ideals, or, in the case of integral domains, by the quotient field.
		Rings associated to group a group G shed light on the structure of G, particular rings of invariants k(V)^G (given a group action on a vector space V), cohomology rings H^(G,Z), group rings* Z[G], and representation rings R(G).
	www.math.niu.edu /~rusin/known-math/index/13-XX.html (2760 words)

	[No title]
		The opposite of a Zariski category is a strict spatial analytic geometry, whose analytic topology coincides with the Zariski topology defined by Diers.
		(h) The opposite of the category of commutative rings is a Zariski geometry; its analytic topology is the Zariski topology.
		Luo ---------------------------------------------------------------- The opposite RING^op of the category RING of commutative rings (with unit) is an analytic category, which is equivalent to the category of affine schemes.
	www.mta.ca /~cat-dist/catlist/1999/abstralggeo (7238 words)

	Kestrel Institute - Research Staff - Dusko Pavlovic - Semantics of computation
		In the present paper, we describe a category of processes modulo strong bisimulations, with the bisimilarity preserving simulations as morphisms, and show that it is equivalent to the category of labelled irredundant trees and the label preserving tree morphisms.
		An abstract construction of a category of processes in a general setting is presented in the appendix.
		We further discuss categories of resumptions and of hyperfunctions, which are the main examples of prcess categories.
	www.kestrel.edu /home/people/pavlovic/semantics.html (1008 words)

	Rings Of Quotients Of f-Rings By Gabriel Filters of Ideals (ResearchIndex)
		It is shown that for every 2--convex semiprime f--ring A and every multiplicative filter G of dense ideals the ring of quotients of A by G, namely the direct limit of the HomA(I;A) over all I 2 G, is an `--subring of QA, the maximum ring of quotients.
		Relative to the category of all commutative rings with identity, it is shown that for every 2-- convex semiprime f--ring A, qA, the classical...
		1 Lattice--ordered rings of quotients (context) - Anderson - 1965
	citeseer.ist.psu.edu /476597.html (625 words)

	Sheaf
		Indeed, the F(U) are commutative rings and the restriction maps are ring homomorphisms, and F is therefore even a sheaf of rings on X.
		If C is also a complete category, then we can extend F to a functor from T to C.
		For some sheaves, germs behave well, and can give good local information; the germ of an analytic function around a point determines the function in a small neighboorhood of the point, using its power series expansion.
	www.brainyencyclopedia.com /encyclopedia/s/sh/sheaf.html (2661 words)

	@CAT 2002-2003
		In the special case of rings satisfying the identity x^2=0 this is indeed so and is proved by a relatively straightforward translation of the group theoty arguement.
		Abstract: Although the inclusion into a commutative ring of its ring of idempotents is not a ring homomorphism, it does preserve the elementary logic operations.
		This suggests an extension of the category of commutative rings to a category of commutative rings and logical morphisms.
	www.mscs.dal.ca /~pare/Sem02-03.html (1493 words)

	[No title]
		Theorem 2.5 shows that the category of B -comodules is equivalent to the local* iza- tion of the category* of -comodules with respect to some hereditary torsion the* *ory T.
		For rings R and S, we can have equivalences of categories between R-modules and S-modules that are not induced by maps R -!S; this is, of course, the conte* *nt of Morita theory.
		Then the category of graded B -comodules is equivalent to the category of graded B* 0- comodules, and both categories* are equivalent to the localizationLof the catego* *ry of graded -comodules with respect to the torsion theory pTh(p)tpB.
	hopf.math.purdue.edu /Hovey-Strickland/torsion-comod.txt (8546 words)

	SHEAF FACTS AND INFORMATION (Site not responding. Last check: 2007-11-06)
		Suppose ''X'' is a topological space, and C is a category (often, this is the category of sets, the category of Abelian_groups, the category of commutative_rings, or the category of modules over a fixed ring).
		Ringed_spaces are sheaves of commutative rings; especially important are the locally_ringed_spaces where all stalks (see below) are local_rings.
		'' is an object of C, for C a category such as the category_of_abelian_groups or the category_of_commutative_rings.
	www.witwib.com /sheaf (2804 words)

	Articles - Morphism (Site not responding. Last check: 2007-11-06)
		A category C is given by two pieces of data: a class of objects and a class of morphisms.
		Epimorphisms in concrete categories are typically surjective morphisms, although this is not always the case.
		In the concrete categories studied in universal algebra (such as those of groups, rings, modules, etc.), morphisms are called homomorphisms.
	oldion.com /articles/Morphism (891 words)

	Sheaf (Site not responding. Last check: 2007-11-06)
		The first is the concept of presheaf, which formalizes the idea of restriction, and can be formulated in terms of elementary category theory.
		Suppose X is a topological space, and C is a concrete category (think of the examples we already encountered above: the category of sets, the category of commutative rings or the category of real vector spaces).
		In the language of category theory, all of this can be summarized as follows: a presheaf of C on X is a contravariant functor from the category of open subsets of X, with inclusions as morphisms, to C.
	www.portaljuice.com /sheaf.html (1959 words)

	MathGuide - Simple Search (Site not responding. Last check: 2007-11-06)
		Field theory and polynomials; Commutative rings and algebras; Algebraic geometry; Linear and multilinear algebra, matrix theory; Associative rings and algebras; Nonassociative rings and algebras; Category theory, homological algebra
		Commutative rings and algebras; Nonassociative rings and algebras; Algebraic geometry; Combinatorics; Algebraic topology
		Commutative rings and algebras; Differential geometry; Functional analysis; Mathematical logic and foundations
	www.mathguide.de /cgi-bin/ssgfi/suche.pl?db=math&tag=SUC&words=13-XX&sort=&dsp=minitemp&COL=SUB (101 words)

	Spectrum
		In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all prime ideals of R.
		It is useful to use the language of category theory and observe that Spec is a functor.
		The functor Spec yields a contravariant equivalence between the category of commutative rings and the category of affine schemes.
	www.websters-online-dictionary.com /definition/english/Sp/Spectrum.html (7719 words)

	[No title]
		Actually, as applied to categories, "left exact" is a thrice-dead metaphor (twice-dead as applied to functors, since "exact sequence" is a dead metaphor for "exact differential", and "left exact" as applied to functors is a dead metaphor for "preserving the left- hand ends of exact sequences").
		Another example of a category which is distributive but not extensive is the dual of the category of unital rings; note that the dual of the category of unital commutative rings is even extensive.
		A fundamental example of a linear "category" is the 2-category of all categories with coproducts; it has an obvious abstraction functor to the linear category of commutative monoids, by taking isomorphism- classes.
	www.mta.ca /~cat-dist/catlist/1999/extensive (7570 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		It is well-known that a Hopf algebroid is the same thing as a presheaf of groupoids on Aff, the opposite category of commutative rings.
		We prove the general theorem that internal equivalences of presheaves of groupoids with respect to a Grothendieck topology on Aff give rise to equivalences of categories of sheaves in that topology.
		The corresponding statement for Hopf algebroids is that weakly equivalent Hopf algebroids have equivalent categories of comodules.
	hopf.math.purdue.edu /Hovey/hopfalgebroids.abstract (177 words)

	Seminários de TEORIA DAS CATEGORIAS (Site not responding. Last check: 2007-11-06)
		Janelidze has defined a very general notion of Galois theory, where the basic data consist of a category C, a reflective full subcategory X, and a class S of the maps in C, with these data subject to mild assumptions.
		The central point of Janelidze's general Galois theory is a description of the coverings of B - or rather of those made trivial by pullback along a GIVEN effective descent map p: E --> B - as the actions on X of a certain Galois pregroupoid of the extension (E,p).
		It is often, but not always, the case that the category of coverings of B is reflective in the category of all S-maps into B. The aim of the present talk is to give sufficient conditions for this, and to prove them satisfied in each of the important examples above.
	www.mat.uc.pt /~categ/seminars/1997/23Jan97.html (158 words)

	Novedad
		The symbol C(X) refers to the ring of real valued functions on a (completely regular) topological space X, a reduced commutative ring.
		However, C(X) lives naturally in many categories and the answer can depend on the choice.
		We look mostly at the category CR of commutative rings and the category R/N of reduced commutative rings.
	www.um.es /matematicas/novedades/2004.03.05.burgess1.html (150 words)

	The Math Forum - Math Library - Rings/Ideals (Site not responding. Last check: 2007-11-06)
		Commutative rings are sets like the set of integers, allowing addition and (commutative) multiplication.
		Of particular interest are several classes of rings of interest in number theory, field theory, and related areas; however, other classes of rings arise, and a rich structure theory arises to analyze commutative rings in general, using the concepts of ideals, localizations, and homological algebra.
		One of its main research interests lies in near-rings: generalised rings that might generally be described as rings (N,+,*) where the addition is not necessarily abelian and only one distributive law...more>>
	mathforum.org /library/topics/rings_ideals (670 words)

	[No title]
		An important example of a model category is the category of unbounded chain complexes of R-modules, which has as its homotopy category the derived category of the ring R. This example shows that traditional homological algebra is encompassed by Quillen's homotopical algebra.
		Examples include the "pure derived category" of a ring R, and derived categories capturing relative situations, including the projective class for Hochschild homology and cohomology.
		Finally, we explain how the category of simplicial objects in a possibly non-abelian category can be equipped with a model category structure reflecting a given projective class, and give examples that include equivariant homotopy theory and bounded below derived categories.
	www.lehigh.edu /dmd1/public/www-data/h64.txt (1226 words)

	[No title]
		to abelian groups; we often consider presheaves with values in different categories, such as the category of sets, or the category of commutative rings.
		Here the restriction maps are the usual restrictions of a function from one set to a smaller set contained inside it.
		with the usual topologies, then the presheaf defined in the previous example takes its values in the category of commutative rings.
	odin.mdacc.tmc.edu /~krc/agathos/sheaf.html (579 words)

	Duality and rational modules in Hopf algebras over commutative rings (ResearchIndex)
		This applies in particular for the canonical pairings (C, C #) and (A #, A) derived from a coalgebra C and an algebra A, respectively.
		An attempt to develop systematically the theory of rational modules associated to a pairing (C, A), where C is a coalgebra and A is an algebra over an arbitrary commutative ring R, is [4].
		A corollary of the theory there developed is that if C is projective as an R--module, then the category of right C--comodules is isomorphic to...
	citeseer.ist.psu.edu /479032.html (366 words)

	OUP: Category Theory 1991: Seely
		As category theory approaches its first half-century, it continues to grow, finding new applications in areas that would have seemed inconceivable a generation ago, as well as in more traditional areas.
		The language, ideas, and techniques of category theory are well suited to discovering unifying structures in apparently different contexts.
		The specification in this catalogue, including without limitation price, format, extent, number of illustrations, and month of publication, was as accurate as possible at the time the catalogue was compiled.
	www.oup.co.uk /isbn/0-8218-6018-6 (296 words)

	Commutative Algebra Seminar
		Abstract: Based on the Gabber and Ramero's almost ring theory I will discuss the class of rings in mixed characteristic for which the Frobenius map is surjective modulo $p$.
		Abstract: Lyubeznik recently proved a criterion for the connectedness of the punctured spectrum of a local ring of positive characteristic.
		The main focus is to discuss the finiteness condition of local cohomology modules of a certain non-Noetherian ring.
	www.math.utah.edu /~spiroff/seminar (763 words)

	Arithmetical Properties Of Commutative Rings And Monoids (Site not responding. Last check: 2007-11-06)
		Book Description: ------------------Description-------------------- The study of nonunique factorizations of elements into irreducible elements in commutative rings and monoids has emerged as an independent area of research only over the last 30 years and has enjoyed a recent flurry of activity and advancement.
		The first seven chapters demonstrate the diversity of approaches taken in studying nonunique factorizations and serve both as an introduction to factorization theory and as a survey of current trends and results.
		The remaining chapters reflect research motivated by arithmetical properties of commutative rings and monoids.
	isbn.nu /0824723279 (416 words)

	On some properties of pure morphisms of commutative rings (Site not responding. Last check: 2007-11-06)
		On some properties of pure morphisms of commutative rings
		We prove that pure morphisms of commutative rings are effective $A$-descent morphisms where $A$ is a (COMMUTATIVE RINGS)$^op$-indexed category given by (i) finitely generated modules, or (ii) flat modules, or (iii) finitely generated flat modules, or (iv) finitely generated projective modules.
		Keywords: Indexed categories, effective descent morphisms, pure morphisms.
	www.tac.mta.ca /tac/volumes/10/9/10-09abs.html (67 words)

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