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	Dimension - Wikipedia, the free encyclopedia
		In mathematics, dimensions are the parameters required to describe the position and relevant characteristics of any object within a conceptual space —where the dimensions of a space are the total number of different parameters used for all possible objects considered in the model.
		The inductive dimension of a topological space may refer to the small inductive dimension or the large inductive dimension, and is based on the analogy that n+1-dimensional balls have n dimensional boundaries, permitting an inductive definition based on the dimension of the boundaries of open sets.
		The Krull dimension of a commutative ring, named after Wolfgang Krull (1899 - 1971), is defined to be the maximal number of strict inclusions in an increasing chain of prime ideals in the ring.
	en.wikipedia.org /wiki/Dimension (1640 words)

	Dimension
		Dimension (from Latin "measured out") is, in essence, the number of degrees of freedom available for movement in a space.
		Time is frequently referred to as the "fourth dimension"; time is not the fourth dimension of space, but rather of spacetime.
		The Krull dimension of a commutative ring is defined to be the maximal length of a strictly increasing chain of prime ideals in the ring.
	www.ebroadcast.com.au /lookup/encyclopedia/2d/2d.html (439 words)

	Krull dimension - Wikipedia, the free encyclopedia
		In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull (1899 - 1971), is defined to be the number of strict inclusions in a maximal chain of prime ideals.
		The Krull dimension is the supremum of the lengths of chains of prime ideals.
		An alternate way of phrasing this definition is to say that the Krull dimension of R is the largest height of any prime ideal of R.
	en.wikipedia.org /wiki/Krull_dimension (265 words)

	Dimension - Facts, Information, and Encyclopedia Reference article
		In common usage, the dimensions of an object are the measurements that define its shape and size.
		The theory of manifolds, in the field of geometric topology, is characterised by the way dimensions 1 and 2 are relatively elementary, the high-dimensional cases n > 4 are simplified by having extra space in which to 'work'; and the cases n = 3 and 4 are in some senses the most difficult.
		Science fiction texts often mention the concept of dimension, when really referring to parallel universes, alternate universes, or other planes of existence, or concepts that are beyond the reader.
	www.startsurfing.com /encyclopedia/d/i/m/Dimension.html (1161 words)

	PlanetMath: dimension
		The word dimension in mathematics has many definitions, but all of them are trying to quantify our intuition that, for example, a sheet of paper has somehow one less dimension than a stack of papers.
		One common way to define dimension is through some notion of a number of independent quantities needed to describe an element of an object.
		This is version 7 of dimension, born on 2003-10-18, modified 2004-03-29.
	planetmath.org /encyclopedia/Dimension3.html (513 words)

	Springer Online Reference Works
		Another, inductive, approach (see Inductive dimension) to the definition of the dimension of a topological space is possible, based on the separation of the space by subspaces of smaller dimension.
		Dimension theory is most meaningful, first, for the class of metric spaces with a countable base, and, secondly, for the class of all metric spaces.
		Several of the dimensions most used in algebra and ring theory may be defined on the lattice of submodules of some module, globalizing the definition by considering the supremum (or a similar invariant) of the dimension of all modules (perhaps restricting to a certain class of modules).
	eom.springer.de /D/d032450.htm (2461 words)

	what does the dimension mean?
		An easy example of the dimension of a 'more obvious' vector space is the dimension of the vector space of all radius-vectors in 3-d Euclidean space.
		dimension is often defined inductively, based on the idea that the boundary of something has dimension one less than the something, boundary in the sense of boundary of a manifold, not topological limit points.
		in algebra and algebraic geometry, dimension of a domain is defined in terms of "krull dimension", i.e.
	www.physicsforums.com /showthread.php?t=80811 (2553 words)

	Dimension - Questionz.net , answers to all your questions (Site not responding. Last check: 2007-10-30)
		Lebesgue covering dimension For any topological space, the Lebesgue covering dimension is defined to be n if any open cover has a refinement (a second cover where each element is a subset of an element in the first cover) such that no point is included in more than n+1 elements.
		Hausdorff dimension For sets which are of a complicated structure, especially fractals, the Hausdorff dimension is useful.
		Krull dimension of commutative rings The Krull dimension of a commutative ring is defined to be the maximal length of a strictly increasing chain of prime ideals in the ring.
	www.questionz.net /Video_games/2d.html (605 words)

	2D
		It is conventional (and for most practical purposes entirely sensible) to consider this as three spatial dimensions and one of time.
		It is somewhat different to the three spatial dimensions in that there is only one of it, and movement seems to be possible in only one direction.
		The concept is used to infer that if one travels into 'another dimension' they are traveling beyond the bounds of human understanding.
	www.comicscomics.com /search.php?title=2D (811 words)

	sze information,size (Site not responding. Last check: 2007-10-30)
		(Incommon usage, the dimensions of an object are the measurements that defineits shape and sze.
		The theory of manifolds, in the field of geometric topology,is characterised by the way dimensions 1 and 2 are relatively elementary, the high-dimensional cases n> 4 are simplified by having extra space in which to 'work'; and the cases n = 3 and 4 are in some senses the mostdifficult.
		The Krull dimension of a commutative ring, named after Wolfgang Krull (1899 - 1971), is defined to be the maximal number of strict inclusions in an increasingchain of prime ideals in the ring.
	www.vsearchmedia.com /sze.html (608 words)

	PlanetMath: Krull dimension
		bound on the Krull dimension of polynomial rings
		Cross-references: subsets, closed, irreducible, topological space, length, prime ideals, sequence, integers, supremum, dimension, ring
		This is version 3 of Krull dimension, born on 2001-12-20, modified 2003-08-01.
	planetmath.org /encyclopedia/DimensionKrull.html (82 words)

	Liens vers les maths constructives
		Dimension de Heitmann des treillis distributifs et des anneaux commutatifs.
		We give an elementary characterization of Krull dimension of commutative rings and we deduce a short proof for the Krull dimension of a polynomial ring over a field.
		Hidden constructions in abstract algebra (3) Krull dimension of distributive lattices and commutative rings fichier poscript fichier pdf, fichier dvi.
	hlombardi.free.fr /liens/constr.html (679 words)

	Research Statement: (Site not responding. Last check: 2007-10-30)
		Then the intuition is that a ``curve category'' is the full subcategory created from a curve module; this notion does seem to be a reasonable analogue to the case of a curve in a larger (commutative) variety.
		C created from a general multistrand module, the 1-critical curve modules of projective dimension 1 are often distinct from those of injective dimension 1.
		dimensions not constant among all 1-critical projective modules; however, injective dimension is constant among the 1-critical projective modules in GrMod(S) for a typical graded ring S. I plan to continue creating this type of example in order to determine the connections between hypotheses placed on I and the conditions satisfied by the associated curve category.
	acad.udallas.edu /mathdept/retert/researchstat.html (2761 words)

	No Title (Site not responding. Last check: 2007-10-30)
		The idea of ``dimension'' is fundamental in many parts of Mathematics.
		Very intuitively, each kind of dimension ``takes the measure'' of the involved concepts from Mathematics in the form of numerical, cardinal, or ordinal invariants.
		The aim of this talk is to present to a general audience some recent results involving various aspects of the Krull dimension, which besides the (co)homological dimension, Goldie dimension, Gabriel dimension, and Gelfand-Kirillov dimension is one of the most important ``dimensions'' encountering in Algebra, and in particular in Ring and Module Theory.
	www.uwm.edu /~gb/COLLOQUIA/99-02-12/99-02-12.html (122 words)

	Dimension : 2d
		Time is frequently referred to as the "fourth dimension"; time isn't the fourth dimension of space, but rather of spacetime.
		This doesn't have a Euclidean geometry, so temporal directions are not entirely equivalent to spatial dimensions.
		Cordery to take her will have us to come and see her before he do go.
	www.findword.org /2d/2d.html (508 words)

	[No title]
		A study of the relative dual Krull dimension of modules and its link with the relative Krull dimension of the underlying ring, cf.
		A study of the relationship between the Krull dimension and the dual Krull dimension of a ring, cf.
		Global Krull dimension, in ``Ring Theory and Representations of Algebras'', Proceedings of the Euroconference Interactions between Ring Theory and Representations of Algebras, University of Murcia, January 1998, edited by M. Saorin and F. Van Oystaeyen, Marcel Dekker, Inc., New York Basel, p.
	gta.math.unibuc.ro /pages/talbu.html (3588 words)

	ARTIN (Site not responding. Last check: 2007-10-30)
		A (Diffusion) Algebra whose Krull Dimension exceeds its Gelfand-Kirillov Dimension
		In a noetherian ring, the gelfand-kirillov dimension is always greater than or equal to the krull dimension.
		I will hopefully present a non-noetherian algebra and convince you that it has krull dimension 4 and gelfand-kirillov dimension 3.
	www.maths.gla.ac.uk /~pt/ARTIN_Oabstract.htm (82 words)

	[No title]
		Hilbert dimension is at least the Hilbert dimension of a quotient.
		Hilbert dimension of a domain exceeds the Hilbert dimension of the quotient by a nonzero element of the maximal ideal.
		Comments on generalizations: the dimension formula; affine domains are catenary.
	cr.yp.to /1998-515/inclass.html (1124 words)

	commalg.org - the center for commutative algebra (Site not responding. Last check: 2007-10-30)
		I describe the role of various concepts from commutative algebra, including finite generation, Krull dimension, depth, associated primes, the Cohen-Macaulay and Gorenstein conditions, local cohomology, Grothendieck's local duality, and Castelnuovo-Mumford regularity.
		Abstract: A study of the relation between a noetherian local domain with a given valuation and its associated graded ring with respect to the valuation, which in some cases is an esentially toric variety, possibly of infinite embedding dimension, but of finite Krull dimension.
		In a more algebraic direction, we show that the number of such jumping coefficients bounds the uniform Artin-Rees number of the principal ideal (f) in the sense of Huneke: in the case of isolated singularities, this in turn leads to bounds involving the Milnor and Tyurina numbers of f.
	www.commalg.org /preprints/2003_03.shtml (2821 words)

	dimension - Wiktionary
		The dimensions of velocity are length divided by time
		When initially tagging an entry with this template, be sure to enclose each language in a {{ttbc...}} tag to subcategorize it properly.
		to dimension (third-person singular simple present dimensions, present participle dimensioning, simple past dimensioned, past participle dimensioned)
	en.wiktionary.org /wiki/dimension (239 words)

	[No title]
		The Yagita invariant p() of is a natural generalization of the p-per* iod to groups with H(; Fp) of Krull dimension larger than one.
		If the Krull dimension of H(g; Fp) equals one so that g is p-periodi c, and if we assume g of the form lpff+ 1 with l prime to p and ff > 0, then the (p)-c *ondition does not hold for g.
		Moreover, ff is then necessar* ily equal to 1 (otherwise the Krull* dimension is larger than 1 by [B]), and we recover th* *e formula p(g) = 2(p - 1) of [GMX] for that case.
	www.math.purdue.edu /research/atopology/Glover-Mislin-Xia/yagita.txt (5974 words)

	Learn more about Dimension in the online encyclopedia. (Site not responding. Last check: 2007-10-30)
		Learn more about Dimension in the online encyclopedia.
		Enter a phrase or search word in the box below.
		Hint: Play with putting spaces before and after your words to see the different results you get.
	www.onlineencyclopedia.org /d/di/dimension.html (669 words)

	Mark Teply: Curriculum Vita
		Ph.D. Koker - "Homological Dimension of Rings with Krull and Gabriel Dimension," August, 1990.
		"On the Transfer of Krull Dimension and Gabriel Dimension to Subidealizers," (written jointly with G. Krause), Canadian J. Math.
		"Dual Krull Dimension and Quotient Finite Dimensionality," (written jointly with T. Albu and M. Iosif), Journal of Algebra, 284 (2005), 52-79.
	www.uwm.edu /~mlteply/cv.html (1710 words)

	Mathematician's Profeesional Homepage, Detail Information
		Gelfand-Kirillov dimension: nilpotency of the radical, representability, non-matrix varieties of algebras.
		Krull and Gabriel dimension of modules over PI-rings and algebras.
		Modules with Krull and Gabriel dimension over rings and algebras with polynomial identities. First International Tainan-Moscow Algebra Workshop, Berlin — New York, 1996, 249-256.
	mech.math.msu.su /~markov/papers.htm (321 words)

	(Site not responding. Last check: 2007-10-30)
		Let (R,m) be a local, Noetherian ring of Krull dimension d, and let M and N be finitely generated R-modules such that M has finite projective dimension and the tensor product of M and N has finite length.
		Included in this is the relevant background material in commutative ring theory and the proofs of the Dimension Inequality for an arbitrary regular local ring and the Nonnegativity, Vanishing and Positivity properties for unramified regular local rings.
		Hilbert-Samuel polynomials, dimension theory of Noetherian rings, normal rings, polynomial rings, the Koszul complex, Cohen-Macaulay modules, homological dimension of Noetherian modules, regular rings, minimal resolutions, positivity of higher Euler-Poincaré characteristics, the multiplicity of a module, the intersection multiplicity of two modules.
	www.math.uiuc.edu /~ssather/TEACH/syllabus_sp02.html (360 words)

	Math Forum Discussions
		I've been reading a bit on the dimension theory and related subjects,
		dimension of Notherian rings should always be finite.
		Exercise 9.6 and the comments following the Krull dimension definition
	mathforum.org /kb/thread.jspa?threadID=96204&messageID=482476 (181 words)

	dimension - OneLook Dictionary Search
		Dimension : Eric Weisstein's World of Mathematics [home, info]
		Phrases that include dimension: fourth dimension, fractal dimension, correlation dimension, dimension stone, fifth dimension, more...
		Words similar to dimension: attribute, dimensional, dimensionality, dimensionally, dimensioned, dimensioning, dimensionless, property, proportion, size, more...
	www.onelook.com /?w=dimension (327 words)

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