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Topic: Ideal theory

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	Ideal (ring theory) - Wikipedia, the free encyclopedia
		In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring.
		Ideals are important because they appear as the kernels of ring homomorphisms and allow one to define factor ring.
		The sum and the intersection of ideals is again an ideal; with these two operations as join and meet, the set of all ideals of a given ring forms a lattice.
	en.wikipedia.org /wiki/Ring_ideal (1224 words)

	Ideal (order theory) - Wikipedia, the free encyclopedia
		The existence of prime ideals is in general not obvious, and often a satisfactory amount of prime ideals cannot be derived within Zermelo-Fraenkel set theory.
		In Stone's representation theorem for Boolean algebras, the maximal ideals (or, equivalently via the negation map, ultrafilters) are used to obtain the set of points of a topological space, whose clopen sets are isomorphic to the original Boolean algebra.
		Ideals were introduced first by Marshall H. Stone, who derived their name from the ring ideals of abstract algebra.
	en.wikipedia.org /wiki/Ideal_(order_theory) (1236 words)

	Ideal (ring theory)
		In ring theory, a branch of abstract algebra, an ideal of a ring R is a subset I of R which is closed under R-linear combinations, in a sense made precise below.
		Ideals are important because they appear as the kernels of ring homomorphisms and allow one to define factor rings, as will be described next.
		The term "ideal" comes from the notion of ideal number: ideals were seen as a generalization of the concept of number.
	www.brainyencyclopedia.com /encyclopedia/i/id/ideal__ring_theory_.html (1418 words)

	Encyclopedia :: encyclopedia : Order theory (Site not responding. Last check: 2007-10-25)
		Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering.
		It addresses readers who have basic knowledge of set theory and arithmetics and who know what a binary relation is, but who are not familiar with order theoretic considerations so far.
		More complicated lower subsets are ideals, which have the additional property that each two of their elements have an upper bound within the ideal.
	www.hallencyclopedia.com /Order_theory (3999 words)

	PlanetMath: ideal
		When the failure of unique factorization in number fields was first noticed, one of the solutions was to work with so-called “ideal numbers” in which unique factorization did hold.
		These “ideal numbers” were in fact ideals, and in Dedekind domains, unique factorization of ideals does indeed hold.
		This is version 12 of ideal, born on 2001-10-19, modified 2006-01-23.
	planetmath.org /encyclopedia/Ideal.html (170 words)

	PlanetMath: ideal
		is a left ideal generated by a single element.
		is an ideal generated by a single element.
		This is version 5 of ideal, born on 2002-10-10, modified 2003-08-29.
	planetmath.org /encyclopedia/Ideal3.html (105 words)

	Will Wilkinson / The Fly Bottle: Libertarian Ideal Theory
		Much libertarian ideal theory proceeds on something like the assumption of a entire society of convinced libertarians (or at least the weaker assumption that it is possible to come to the kind of consensus necessary to install a libertarian constitution or basic structure).
		If libertarian ideals are to become more broadly accepted, it may be in part because of more savvy on the part of libertarians in intentionally undermining widespread collectivist impulses.
		So what we need is a theory of just how libertarian a particular society could possibly get, given human psychology, the set of social and economic relations, the available mechanisms of persuasion, and the set of belief systems or "macro mythologies", at a given time, plus the dynamics that govern changes in these things.
	willwilkinson.net /flybottle/archives/2004/02/libertarian_ide.html (885 words)

	Applying TRIZ and the Theory of Ideal SuperSmart Learning to Computing Systems: Ultimate Ideal Autonomous Objects, ...
		I had described the Theory of Ideal SuperSmart Learning as similar to a “theory of everything for product, personal, business, and institutional development.” The goal of the theory is “to know and understand everything from nothing and in no time.” This goal is based on utopic ideality.
		The Theory of Ideal SuperSmart Learning in combination with TRIZ could be used as a resource for developing the following: new paradigms for computing systems; new thinking about objects; new framings for apparently unsolvable problems; new approaches to organizing ideas for strategic problem solving and innovation.
		The key proposals of this paper are more widespread use of the model of “ultimate ideal autonomous (autonomic) object” for computing systems as well as case study applications of tools of TRIZ and the Theory of Ideal Supersmart Learning in the area of computing.
	www.triz-journal.com /archives/2002/09/e (6319 words)

	Ideal Theory of Orders
		Since the category of associative algebras does not require that an algebra have a unit, ideals are valid objects in the category of associative algebras.
		The ideal theory of definite orders over Z is highly developed, allowing enumeration of all classes of ideals locally free over an order S, which is assumed to be an Eichler order or for which the index in an Eichler order is of valuation at most 1 at each ramified prime of the algebra.
		Given an ideal I, returns the left order of I, defined as the ring of all elements of the quaternion algebra of I mapping I to itself under left multiplication.
	www.umich.edu /~gpcc/scs/magma/text905.htm (873 words)

	Physics and Astronomy Glossary
		One postulate of relativity theory is that the speed of light is an invariant quantity.
		When the theory is expressed in four dimensional form, with the appropriate choice of quantities, new invariant quantities emerge: the world-displacement (x + y + z +ict), the energy-momentum four-vector, and the electric and magnetic potentials may be combined into an invariant four-vector.
		Theories are said to be successful if (1) they synthesize and unify a significant range of phenomena; (2) they have predictive power, either predicting new phenomena, or suggesting a direction for further research and testing.
	www.physlink.com /Reference/Glossary.cfm (8054 words)

	Ideal Theory - Cambridge University Press
		Ideal theory is important not only for the intrinsic interest and purity of its logical structure but because it is a necessary tool in many branches of mathematics.
		In this introduction to the modern theory of ideals, Professor Northcott assumes a sound background of mathematical theory but no previous knowledge of modern algebra.
		After a discussion of elementary ring theory, he deals with the properties of Noetherian rings and the algebraic and analytical theories of local rings.
	www.cambridge.org /catalogue/catalogue.asp?isbn=0521604834 (155 words)

	Newsgroops - Ideal (order theory), wikipedia (Site not responding. Last check: 2007-10-25)
		A subset I of a lattice (P,=<) is a prime ideal, iff
		It is defined to be a proper ideal I with the additional
		ideal that extends the above conditions for infinite meets.
	www.newsgroops.org /group/sci.math/article-546500.html (193 words)

	ABSTRACT ALGEBRA ON LINE: Ideal Theory of Commutative Rings
		An integral domain D is called a Dedekind domain if each proper ideal of D can be written as a product of a finite number of prime ideals of D. We will show in Theorem 12.2.4 that a Dedekind domain has some of the properties of a principal ideal domain.
		I is the intersection of all prime ideals of R that contain I. In any principal ideal domain, our next definitions both reduce to the statement that the ideal in question is generated by a power of an irreducible element.
		One important consequence of the generalized principal ideal theorem is that any Noetherian ring satisfies the descending chain condition for prime ideals.
	www.math.niu.edu /~beachy/aaol/commutative.html (2296 words)

	Justice and Bad Luck
		The underlying idea is that every person's starting point in society is the outcome of a social lottery (the political, social, and economic circumstances into which each person is born) and a natural lottery (the biological potentials each person is born with).
		They undermine alternative theories in which distributions of social and economic benefits deviating from that prescribed by the difference principle are tolerated (Nozick 1974, 216; Arneson 2001, 76).
		In any case, luck-egalitarians are unlikely to claim that luck-neutralization is the only ideal, as that would imply that a world where everyone lives miserable lives is better, all things considered, than a world where half the people live tremendous lives and the other half live even better lives.
	www.seop.leeds.ac.uk /entries/justice-bad-luck (8441 words)

	_:: IDEAL THEORY ::_
		Whatever your design needs are IDEAL THEORY can handle it.
		If you need customized stationery or a website, we will maintain the look and feel of your new design throughout all of your media materials, to ensure that your business reflects a consistent visual identity.
		IDEAL THEORY provides lifetime support for your product.
	www.idealtheory.com /design.htm (192 words)

	_:: IDEAL THEORY ::_
		Since 1999, IDEAL THEORY has been providing quality marketing materials crafted to your needs as well as budget.
		IDEAL THEORY offers high quality service that is backed by high quality standards.
		IDEAL THEORY is just a small part of your business but our presence can be one of the biggest factors in the development of your business future.
	www.idealtheory.com (221 words)

	A generalization model based on OI-implication for ideal theory refinement (Site not responding. Last check: 2007-10-25)
		A framework for theory refinement is presented pursuing the efficiency and effectiveness of learning regarded as a search process.
		Past results have demonstrated the impossibility of specifying ideal operators in search spaces where standard generalization models, like logical implication or q-subsumption, are adopted.
		By assuming the object identity bias over a space defined by a clausal language ordered by logical implication, a novel generalization model, named OI-implication, is derived and we prove that ideal operators can be defined for the resulting search space.
	www.cs.bris.ac.uk /~ILPnet2/Tools/Reports/Abstracts/2001-esposito.html (123 words)

	☞ research online - Ideal (Site not responding. Last check: 2007-10-25)
		Ideal is a leading designer and manufacturer of worm drive and specialty hose clamps for automotive...
		theory, a branch of abstract algebra, an ideal is a special subset of a ring.
		Ideal (ring theory), in mathematics, special subsets of a ring considered in abstract algebra.
	www.research-123.info /ideal (424 words)

	KLUEDO - Algorithmic ideal theory
		In the second chapter we introduce standard bases (generalization of Gröbner bases to non-well-orderings), which are needed for applications to local algebraic geometry (singularity theory), and a method for computing syzygies and free resolutions.
		In the third chapter several algorithms for primary decomposition of polynomial ideals are presented, together with a discussion of improvements and preferable choices.
		The last chapter gives an elementary introduction to singularity theory and then describes algorithms, using standard bases, to compute infinitesimal deformations and obstructions, which are basic for the deformation theory of isolated singularities.
	kluedo.ub.uni-kl.de /volltexte/2000/794 (293 words)

	Kinetic Theory and Ideal Gas
		Those balloons are a sort of Ideal Gas thermometer.
		Ideality is a tad easier to believe even though it is only arrived at in a truly tenuous limit.
		Finally, we need to calculate the pressure P as a summation of molecular collisional forces, F, on area, A, of the container as P=F/A. Newton assures us that those forces arise by transfer of molecular momentum, p (not P), in elastic collisions with the wall.
	www.utdallas.edu /~parr/chm5414/54140905.html (1329 words)

	ideal theory
		Scientists who are considered to be great developed the ideal theory.
		Of course what I meant (or more appropriately want to mean) is that the scientists who developed ideal theory are indeed great but I have a misunderstanding about their observations due to my misunderstanding.
		Indeed on a theoretical point of view, ideal theory is the milestone of many theories in statistical mechanics, i.e.
	www.physicsforums.com /showthread.php?t=99239 (987 words)

	Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations
		Numerical codes based on the ideal flow theory have been mainly developed for nonsteady sheet (membrane) metal forming processes.
		The simplest theory of this class is the theory of plane strain deformation.
		In general, the ideal flow theory is restricted to perfectly plastic materials.
	www.osti.gov /energycitations/product.biblio.jsp?osti_id=20558180 (362 words)

	Basic Algorithms in Ideal Theory (Site not responding. Last check: 2007-10-25)
		Gröbner and local standard bases are the heart of several basic algorithms in ideal theory, see e.g.
		), the quotient of a module by an ideal, and the stable quotient of a module by an ideal.
		Algorithms for their computation can be obtained from the corresponding algorithms for a single element as divisor either by the generic element method [8] or as an intersection [6.31]BKW.
	www.uni-koeln.de /REDUCE/cali/node24.html (173 words)

	ideal gas theory
		I think the only chance of disproving ideal theory is experiment.
		You start by simply asserting that the "ideal gas law" must be wrong and then conclude that scientists who did experiments leading to that law must have done them wrong.
		The ideal gas law is a good approximation for many real gasses.
	sci4um.com /about15429.html (760 words)

	Mathematical Theory For Ideal Shape in Nature
		Short and his colleagues have developed a mathematical theory that explains how stalactites get their shape.
		"It's an ideal shape in nature and in mathematics that had not been known before," said Raymond Goldstein, a UA physics professor and senior author on the research report.
		Back in the lab, the researchers analyzed the actual stalactites and compared their shapes to the ideal form predicted by the mathematics.
	www.physorg.com /news2244.html (1099 words)

	Ideal Theory and Gröbner Bases (New) [HB 50] (Site not responding. Last check: 2007-10-25)
		Ideal Theory and Gröbner Bases (New) [HB 50]
		Many of the standard functions based on Grobner bases over fields also carry over to ideals defined over Euclidean rings.
		Ideals may be defined over general Euclidean rings, and Gröbner bases of such ideals may be computed.
	www.math.niu.edu /help/math/magmahelp/rel/node40.html (202 words)

	Algorithmic Ideal Theory (ResearchIndex) (Site not responding. Last check: 2007-10-25)
		Abstract: this paper that he proved the "division theorem by an ideal".
		5 Advances and improvements in the theory of standard bases an..
		1 Initial ideals of Veronese subrings (context) - Eisenbud, Reeves et al.
	citeseer.ist.psu.edu /141578.html (992 words)

	ideal - Definitions from Dictionary.com
		: a standard of perfection, beauty, or excellence ideals are … the essential manifestations of the superego —G. Blum> ideal and not the actual pattern of behavior —Weston La Barre>
		Bosco and Elio Giovannetti combining Miranda and Prolog.
		In domain theory, a non-empty, downward closed
	dictionary.reference.com /search?q=ideal (269 words)

	Kinetic Theory of Ideal Gases (Site not responding. Last check: 2007-10-25)
		An ideal gas is a gas where the atoms do not exert forces on each other but they do collide with the walls of the container (in elastic collisions).
		Based on common sense and experiment the ideal gas law relates the pressure, temperature, volume, and number of moles of ideal gas:
		If by N we mean the total number of atoms of the gas then we also can write,
	www.unf.edu /~jgarner/kinthy.html (259 words)

	definition of ideal (Site not responding. Last check: 2007-10-25)
		Reaching an imaginary standard of excellence; fit for a model; faultless; as, ideal beauty.
		Teaching the doctrine of idealism; as, the ideal theory or philosophy.
		An, As, Beauty, Conception, Doctrine, Etc, Excellence, Fancy, Fit, For, Idea, Ideal, Idealism, Imaginary, Imagination, In, Intellectual, Knowledge, Mental, Model, Of, Only, Or, Perfection, Philosophy, Standard, Teaching, The, Theory, Thought, Unreal, Visionary
	www.brainydictionary.com /words/id/ideal175677.html (207 words)

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