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Topic: Prime ideal theorem

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Boolean prime ideal theorem

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	Boolean prime ideal theorem - Definition, explanation
		The Boolean prime ideal theorem is the strong prime ideal theorem for Boolean algebras.
		to the MIT and PIT for Boolean algebras).
		This is of practical importance for proving Stone's representation theorem for Boolean algebras, a special case of Stone duality, in which one equips the set of all prime ideals with a certain topology and can indeed regain the original Boolean algebra (up to isomorphism) from this data.
	www.calsky.com /lexikon/en/txt/b/bo/boolean_prime_ideal_theorem.php (1671 words)

	Boolean prime ideal theorem - Wikipedia, the free encyclopedia
		The Boolean prime ideal theorem is the strong prime ideal theorem for Boolean algebras.
		to the MIT and PIT for Boolean algebras).
		This is of practical importance for proving Stone's representation theorem for Boolean algebras, a special case of Stone duality, in which one equips the set of all prime ideals with a certain topology and can indeed regain the original Boolean algebra (up to isomorphism) from this data.
	en.wikipedia.org /wiki/Boolean_prime_ideal_theorem (1797 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)
		An ideal can be used to construct a factor ring in a similar way as a normal subgroup in group theory can be used to construct a factor group.
		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.
		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)

	Prime ideal - Definition, explanation
		In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers.
		Prime ideals in order theory are treated in the article on ideals in order theory.
		One use of prime ideals occurs in algebraic geometry, where varieties are defined as the zero sets of ideals in polynomial rings.
	www.calsky.com /lexikon/en/txt/p/pr/prime_ideal.php (788 words)

	PlanetMath: every prime ideal is radical
		"every prime ideal is radical" is owned by alozano.
		Cross-references: induction hypothesis, clear, proposition, induction, inclusion, radical ideal, prime ideal, commutative ring
		This is version 2 of every prime ideal is radical, born on 2003-09-08, modified 2003-09-08.
	www.planetmath.org /encyclopedia/EveryPrimeIdealIsRadical.html (119 words)

	Springer Online Reference Works
		This relation between the points of the interval and the maximal ideals has resulted in the construction of various theories for representing rings as rings of functions on a topological space.
		In a distributive lattice, as in a commutative ring, all maximal ideals are prime; the converse implication holds in a Boolean algebra, and indeed a distributive lattice in which all prime ideals are maximal is necessarily Boolean.
		The construction of maximal ideals in arbitrary rings or lattices generally requires an appeal to Zorn's lemma (see Axiom of choice or Zorn lemma), and indeed the maximal ideal theorem for many classes of rings or lattices (i.e.
	eom.springer.de /M/m062960.htm (555 words)

	Axiom of choice - Wikipedia, the free encyclopedia
		A third possibility is to prove theorems using neither the axiom of choice nor its negation, a tactic preferred in constructive mathematics.
		Important theorems equivalent to AC There are a remarkable number of important statements that, assuming the axioms of ZF but neither AC nor ¬AC, are equivalent to the axiom of choice.
		The Vitali theorem on the existence of non-measurable sets which states that there is a subset of the real numbers that is not Lebesgue measurable.
	en.wikipedia.org /wiki/Axiom_of_choice (3385 words)

	PlanetMath: Birkhoff prime ideal theorem
		"Birkhoff prime ideal theorem" is owned by CWoo.
		Cross-references: join, finite, iff, complete, contradiction, right, side, distributive, meet, contain, ideal generated by, maximal element, Zorn's lemma, upper bound, definition, ideals, chain, poset, inclusion, order, prime, prime ideal, lattice ideal, distributive lattice
		This is version 4 of Birkhoff prime ideal theorem, born on 2007-05-03, modified 2007-05-03.
	planetmath.org /encyclopedia/BirkhoffPrimeIdealTheorem.html (159 words)

	Math Forum Discussions - Prerequisites for Hartshorne's Algebraic Geometry (ver. 1.0) [16]
		Let P be a prime ideal of A. Let S = f(A - P).
		A_P -> B[1/S] Suppose P = QA for a prime ideal Q of B. Since Q does not meet S, QB[1/S] is a prime ideal of B[1/S] by (theorem 20, [7]).
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	www.mathforum.com /kb/thread.jspa?forumID=13&threadID=77315&messageID=355091 (883 words)

	Prime Ideal Correspondence (Site not responding. Last check: 2007-10-29)
		This is similar to the ideal correspondence theorem, but now we are dealing with prime ideals.
		Therefore an ideal is prime iff its image or preimage is prime.
		The ideal p, generated by y, is no longer prime, since it contains x×x, and x is not in p.
	www.mathreference.com /ring,primcor.html (203 words)

	[No title]
		Bergman observes that Theorem 13 applies to inverting the set S of atoms in any Bezout domain R -1 thus a given Bezout domain R ties in our class of rings if and only if the same is true for the ring K obtained from R by inverting the atoms.
		The conditions (l)-(4) for P to be a prime matrix ideal are readily verified using (35), (36), and the "transpose" of (36).
		RpIP consists of all 1 x I matrices of r-rank 0, and by (33) this is the ideal generated by the entries of the idempotent matrices E with rE = 0.
	www.math.rutgers.edu /~sontag/FTP_DIR/wdicks.txt (11071 words)

	Robert Cowen's Page
		A characterization of logical consequence in quantification theory, (pdf file) Notre Dame J. Formal Logic 16, 374-377.
		A compactness theorem for infinite constraint satisfaction, Reports on Math.
		Graph coloring compactness theorems equivalent to BPI (with S.H.Hechler and P.Mihok), Scientiae Mathematicae Japonicae, 56, no.2, 213-223 (pdf file).
	home.nyc.rr.com /rcowen (504 words)

	Schechter: papers
		Kelley's specialization of Tychonov's Theorem is equivalent to the Boolean Prime Ideal Theorem.
		We construct a family of Banach spaces whose bounded sets are precisely the subsets of KH[0,1] that are equiintegrable and pointwise bounded.
		Abstract: Brouwer's Fixed Point Theorem and related theorems (Schauder, Kakutani, etc.) are proved using a topological argument (e.g., compactness) together with a combinatorial argument (e.g., Sperner's Lemma).
	www.math.vanderbilt.edu /~schectex/papers (1198 words)

	Topological Equivalents of the Axiom of Choice and of Weak Forms of Choice, by Eric Schechter
		The Boolean Prime Ideal Theorem and the Axiom of Dependent Choice, discussed briefly below, are known to be strictly weaker than the Axiom of Choice.
		Although the term ``constructive'' is used in different fashion by different mathematicians, the Axiom of Dependent Choice is the strongest form of choice that is widely held to be constructive.
		The web page includes some additional information which may be of interest: lists of equivalents of AC, PI, DC, and HB (the Hahn-Banach Theorem, another weak form of choice), and a chart showing the relations between some of the weak forms of choice.
	at.yorku.ca /z/a/a/b/18.htm (848 words)

	On the prime ideal theorem and irregularities in the distribution of primes, M. Nair, A. Perelli
		On the prime ideal theorem and irregularities in the distribution of primes, M. Nair, A. Perelli
		On the prime ideal theorem and irregularities in the distribution of primes
		On the expression of a number as a sum of primes, Acta Math.
	projecteuclid.org /Dienst/UI/1.0/Display/euclid.dmj/1077286144 (261 words)

	Reports on Mathematical Logic (Site not responding. Last check: 2007-10-29)
		Certain restricted versions of CT, the Compactness Theorem for propositional logic, are introduced by imposing additional conditions on the number of occurrences of propositional variables in sets of formulas.
		In addition we also consider corresponding versions of some equivalents of CT. The introduced versions turn out to be neither provable in ZF set theory nor equivalent to BPI, the Boolean Prime Ideal theorem.
		The restricted versions of CT were suggested by Robert Cowen in connection with his work: Two hypergraph theorems equivalent to BPI, Notre Dame Journal of Formal Logic 31(1990), 232-240, on intimate relations between BPI and NP - complete decision problems.
	www.iphils.uj.edu.pl /rml/rml-25/a-kol-25.htm (128 words)

	Atlas: The general prime ideal theorem by Jan Paseka (Site not responding. Last check: 2007-10-29)
		A useful separation lemma for partial cm-lattices is proved equivalent to PIT, the Prime Ideal Theorem.
		The General Prime Ideal Theorem is formulated and proved to be equivalent to the PIT.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # capw-52.
	atlas-conferences.com /c/a/p/w/52.htm (101 words)

	ABSTRACT ALGEBRA ON LINE: Contents
		This site contains many of the definitions and theorems from the area of mathematics generally called abstract algebra.
		It is intended for undergraduate students taking an abstract algebra class at the junior/senior level, as well as for students taking their first graduate algebra course.
		Ideals in the localization of an integral domain(5.8.11)
	www.math.niu.edu /~beachy/aaol/contents.html (401 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		In all the resources I have available, Deligne's Theorem (axiom) is proven to be a consequence of axiom of choice.
		Am I wrong to think it is actually a consequence of only prime ideal theorem???
		The best possible result I can imagine is that *constructively* every coherent topos E admits a surjection sh(X) ---> E (ie epimorphism in the category of geometric morphisms) where X is a coherent locale.
	www.mta.ca /~cat-dist/catlist/1999/coherent-toposes (100 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		The completeness theorem for coherent propositional logic, alias the Prime Ideal Theorem (PIT) for distributive lattices, is true in certain non-boolean toposes.
		We introduce an extension of coherent propositional logic whose completeness theorem is stronger than (PIT).
		In particular, it is strong enough to conclude (OEP) for arbitrary objects.
	www.math.mcgill.ca /rags/seminar/egger.a18.html (85 words)

	Quine's New Foundations
		At all events, we have to be careful to keep to a minimum the number of sinews we cut, for there are important theorems (Gödel's Incompleteness theorem among others) whose proofs rely on the same “self-reference” that we find in the paradoxes and which we are loath to abandon.
		There is a widespread misapprehension that Cantor's theorem shows that there cannot be a consistent set theory with a universal set.
		That error can be refuted by Jensen's construction of his models for NFU, but by far the most appealing demonstration is a later construction of a model of a set theory with a universal set by Church and Oswald simultaneously and independently in 1974.
	plato.stanford.edu /entries/quine-nf (6401 words)

	AMCA: Smooth Beurling integers from irregular Beurling primes by Hugh Montgomery (Site not responding. Last check: 2007-10-29)
		Roughly a century ago, Landau developed a method for proving the Prime Number Theorem by means of local lemmas.
		In this way, Landau proved the Prime Ideal Theorem for algebraic number fields.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
	at.yorku.ca /c/a/e/w/57.htm (170 words)

	Boolean Prime Ideal Theorem Encyclopedia Articles @ LocalColorArt.com (Local Color Art) (Site not responding. Last check: 2007-10-29)
		Boolean Prime Ideal Theorem Encyclopedia Articles @ LocalColorArt.com (Local Color Art)
		Find More Information about "Boolean prime ideal theorem" in LocalColorArt.com's:
		"Boolean prime ideal theorem" results in these other popular encyclopedia sites:
	216.92.85.60 /encyclopedia/Special:Allpages/Boolean_prime_ideal_theorem (186 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		Dear categories, In ``Stone spaces'', Johnstone notes that the assumption that every coherent locale is spatial is equivalent to the prime ideal theorem for distributive lattices.
		Is it generally agreed that the assumption that the spatial part of every coherent locale has coherent topology is also equivalent to the above two axioms?
		> > Dear categories, > > In ``Stone spaces'', Johnstone notes that the assumption that > (A) every coherent locale is spatial > > is equivalent to > > (B) the prime ideal theorem for distributive lattices.
	www.mta.ca /~cat-dist/catlist/1999/coherent-locales (311 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		On the most general plane closed point set through which it is possible to pass a pseudo-arc
		The independence of the axiom of choice from the Boolean prime ideal theorem
		A remark on Sikorski's extension theorem for homomorphisms in the theory of Boolean algebras
	journals.impan.gov.pl /cgi-bin/shvold?fm55 (100 words)

	Professor John L. Bell
		"The Strength of the Sikorski Extension Theorem for Boolean Algebras", Journal of Symbolic Logic 48, 1983.
		"On the Relationship between the Boolean Prime Ideal Theorem and Two Principles of Functional Analysis", Bull.
		"The Sikorski Extension Theorem for Boolean Algebras", Mathematics Department, McMaster University, February 1991.
	publish.uwo.ca /~jbell (2110 words)

	Robert Cowen (Site not responding. Last check: 2007-10-29)
		I am most interested in the interplay between Logic, Set Theory, Graph Theory and Theoretical Computer Science.
		I have written a number of papers on connections between theorems equivalent to the Boolean Prime Ideal Theorem, and NP-Completeness.
		I have also become interested in using Mathematica in the classroom to involve students in discovering mathematics.
	www.math.qc.edu /web/faculty/cowen.htm (73 words)

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