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Topic: Dedekind domain

Related Topics

Prime ideal

Glossary of ring theory

Unique factorization domain

Division ring

	Kids.Net.Au - Encyclopedia > Dedekind domain
		In other words, a Dedekind domain is a commutative ring which is not a field, doesn't have zero divisors, and in which every ideal is finitely generated, every nonzero prime ideal is a maximal ideal, and which is integrally closed in its fraction field.
		The most important examples of Dedekind domains, and historically the motivating ones, arise from algebraic number fields: start with a finite field extension F of the rational numbers Q and consider the set of all elements of F which are algebraic integers (in other words, the integral closure of Z in F).
		The study of Dedekind domains began when Dedekind introduced the notion of ideal in a ring in the hopes of compensating for the failure of unique factorization into primes in rings of algebraic integers.
	www.kids.net.au /encyclopedia-wiki/de/Dedekind_domain (481 words)

	NationMaster - Encyclopedia: Principal ideal domain (Site not responding. Last check: 2007-10-23)
		In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element).
		In Ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R. The ideal (x,y) generated by x and y, which consists of all the polynomials in C[x,y] that have zero for the constant term, is not principal.
		A principal ideal domain (PID) is an integral domain that is principal.
	www.nationmaster.com /encyclopedia/Principal-ideal-domain (845 words)

	Dedekind domain - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-23)
		In other words, a Dedekind domain is a commutative ring which is not a field, doesn't have zero divisors, and in which every ideal is finitely generated, every nonzero prime ideal is a maximal ideal, and which is integrally closed in its fraction field.
		The most important examples of Dedekind domains, and historically the motivating ones, arise from algebraic number fields: start with a finite field extension F of the rational numbers Q and consider the set of all elements of F which are algebraic integers (in other words, the integral closure of Z in F).
		The study of Dedekind domains began when Dedekind introduced the notion of ideal in a ring in the hopes of compensating for the failure of unique factorization into primes in rings of algebraic integers.
	en.wikipedia.org /wiki/Dedekind_domain (477 words)

	Dedekind cut - Encyclopedia Glossary Meaning Explanation Dedekind cut
		In mathematics, a Dedekind cut in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for any element x in S, if a is in A and x ≤ a, then x is in A as well) and B is closed upwards.
		Regard one Dedekind cut { A, B } as less than another Dedekind cut { C, D } if A is a proper subset of C, or, equivalently D is a proper subset of B.
		The Dedekind cut is named after Richard Dedekind, who invented this construction in order to represent the real numbers as Dedekind cuts of the rational numbers.
	www.encyclopedia-glossary.com /en/Dedekind-cut.html (529 words)

	Dedekind domain -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-23)
		An alternative characterization of Dedekind domains is that an integral domain R is a Dedekind domain if and only if the (A determination of the location of something) localization of R at each prime ideal P of R is a (Click link for more info and facts about discrete valuation ring) discrete valuation ring.
		While not all Dedekind domains are (Click link for more info and facts about unique factorization domains) unique factorization domains, they all have the following property which is in practice often "close enough": every ideal can be uniquely factored as a product of (Click link for more info and facts about prime ideals) prime ideals.
		If R is a Dedekind domain with fraction field E, then a fractional ideal I is an additive ((mathematics) a subset (that is not empty) of a mathematical group) subgroup of E such that RI ⊆ I and such there exists an r in R with rI ⊆ R.
	www.absoluteastronomy.com /encyclopedia/D/De/Dedekind_domain.htm (296 words)

	Principal ideal domain - Wikipedia, the free encyclopedia
		In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element).
		All Euclidean domains are principal ideal domains, but the converse is not true.
		The previous three statements give the definition of a Dedekind domain, and hence every principal ideal domain is a Dedekind domain.
	en.wikipedia.org /wiki/Principal_ideal_domain (306 words)

	Reference.com/Encyclopedia/Dedekind domain
		A Dedekind domain can be defined as an integrally closed domain with Krull dimension at most 1.
		If R is a Dedekind domain with field of fractions K, L/K is a finite field extension, and S is the integral closure of R in L, then S is again a Dedekind domain.
		The study of Dedekind domains began when Richard Dedekind introduced the notion of ideal in a ring in the hopes of compensating for the failure of unique factorization into primes in rings of algebraic integers.
	www.reference.com /browse/wiki/Dedekind_domain (594 words)

	PlanetMath: Dedekind domain
		A Dedekind domain is a commutative integral domain
		It is worth noting that the second clause above implies that the maximal length of a strictly increasing chain of prime ideals is 1, so the Krull dimension of any Dedekind domain is at most 1.
		This is version 13 of Dedekind domain, born on 2002-04-19, modified 2006-04-12.
	planetmath.org /encyclopedia/DedekindDomain.html (170 words)

	PlanetMath: ideal decomposition in Dedekind domain
		According to the entry “fractional ideal”;, we can state that in a Dedekind domain
		"ideal decomposition in Dedekind domain" is owned by pahio.
		This is version 6 of ideal decomposition in Dedekind domain, born on 2005-07-12, modified 2005-07-15.
	www.planetmath.org /encyclopedia/IdealDecompositionInDedekindDomain.html (132 words)

	Principal ideal domain
		The ring Z[X] of all polynomials with integer coefficients is not principal, since for example the ideal generated by 2 and X cannot be generated by a single polynomial.
		Every principal ideal domain is a unique factorization domain (UFD).The converse does not hold since for any field K, K[X,Y] is a UFD but is not a PID (to prove this look at the ideal generated by.
		An example of a principal ideal domain that is not a euclidean domain is the ring (Wilson, J. "A Principal Ring that is Not a Euclidean Ring." Math.
	www.brainyencyclopedia.com /encyclopedia/p/pr/principal_ideal_domain.html (331 words)

	[No title]
		R is a unique factorization domain with a unique irreducible element.
		There is some Dedekind valuation ν on the field of fractions K of R, such that R={x:x in K, ν(x) ≥ 0}.
		Any localization of a Dedekind domain is a discrete valuation ring; in practice, this is frequently how discrete valuation rings arise.
	en-cyclopedia.com /wiki/Discrete_valuation_ring (212 words)

	ABSTRACT ALGEBRA ON LINE: Ideal Theory of Commutative Rings
		Specifically, a Dedekind domain must be Noetherian, and any nonzero prime ideal of a Dedekind domain must be maximal.
		Let D be an integral domain with quotient field F. Assume that D is Noetherian and that every nonzero prime ideal of D is maximal.
		In the setting of the previous theorem, if we assume in addition that R is an integrally closed domain, then a further condition holds, known as ``Going down'': Let R be a subring of the integral domain D, assume that D is an integral extension of R, and that R is an integrally closed domain.
	www.math.niu.edu /~beachy/aaol/commutative.html (2296 words)

	Michiel Hazewinkel : Book review
		Chapter 1 is devoted to algebraic foundations (fields and algebras and the integrality and Noetherian properties), while Chapter 2 (Dedekind domains) gives some essential facts on valuations, completions and module theory over a Dedekind domain, containing the genesis of a whole host of ideas which are basic for the subject.
		Frequently, the study of a suitable ring of algebraic numbers will help in solution of a problem which initially had been stated entirely in terms of ordinary integers: for instance, questions concerning the integral (or rational) coefficients can frequently be dealt with by the study of a suitable ring of algebraic integers.
		The algebraic number theory begins (in Chapter 2) by proving that, if L is a finite separable extension of the field of fractions K of a Dedekind domain d, then D, the integral closure of d in L, is a Dedekind domain.
	homepages.cwi.nl /~mich/reviews/AAA_1255.html (801 words)

	Citations: a Dedekind domain - GL, of (ResearchIndex) (Site not responding. Last check: 2007-10-23)
		Citations: a Dedekind domain - GL, of (ResearchIndex)
		The domain of the modular forms F is in....
		, On GL(n) of a Dedekind domain, Quart.
	citeseer.ist.psu.edu /context/525878/0 (255 words)

	Proceedings of the American Mathematical Society
		Abstract: The following dichotomy is established: A finitely generated, complex Dedekind domain that is not commutative is a simple ring.
		Weaker versions of this dichotomy are proved for Dedekind prime rings and hereditary noetherian prime rings.
		Additional Notes: The research of both authors was partially supported by grants from the National Science Foundation.
	www.ams.org /proc/2005-133-03/S0002-9939-04-07574-4/home.html (326 words)

	Dedekind rings (integral domain, discrete valuation ring).
		The main objective of this paper is to present a brief, yet complete, exposition on a type of integral domain called Dedekind ring.
		In addition, it methodically explores some of the key theorems in relation to the ideals and quotient rings of Dedekind domains to establish the theorem that every ideal in a Dedekind ring can be factored uniquely into prime ideals.
		Also, in the latter part of the paper, a reference is made to the roles that the ideas of relative degree and ramification index play with regard to the factorization of ideals in Dedekind domains.
	escholarship.bc.edu /dissertations/AAI1389162 (160 words)

	Discrete valuation ring (Site not responding. Last check: 2007-10-23)
		R is a local principal ideal domain, and not a field.
		There is some Dedekind valuation ν on the field of fractions K of R, such that R=.
		Note that R is the localization of the Dedekind domain Z at the prime ideal generated by 2.
	www.worldhistory.com /wiki/D/Discrete-valuation-ring.htm (313 words)

	Dedekind Domains (Site not responding. Last check: 2007-10-23)
		A dedekind domain (biography) is a beautiful generalization of a unique factorization domain (ufd).
		A dedekind domain is similar, but now we're working with ideals.
		If sj = 0 then j = 0 (integral domain), which is a proper ideal in h, so set this case aside.
	www.mathreference.com /id-dd,intro.html (408 words)

	Prime ideal - Wikpedia (Site not responding. Last check: 2007-10-23)
		If M is a smooth manifold, R is the ring of smooth functions on M, and x is a point in M, then the set of all smooth functions f with f(x) = 0 forms a prime ideal (even a maximal ideal) in R.
		A commutative ring is an integral domain if and only if {0} is a prime ideal.
		A commutative ring is a field if and only if {0} is its only prime ideal, or equivalently, if and only if {0} is a maximal ideal.
	www.bostoncoop.net /~tpryor/wiki/index.php?title=Prime_ideal (768 words)

	BBC Education - AS Guru - Maths - Pure - Functions - Properties of Functions
		This lesson will discuss a fundamental classification which sets out one group of 'analytic expressions' which cannot be called functions.
		This looks very complicated, but in fact the key issue that we will take account of in this lesson is that Dedekind used the word 'thing' and not 'things'.
		on a given domain is a rule which gives each number
	www.bbc.co.uk /education/asguru/maths/13pure/02functions/09propfunctions/index.shtml (280 words)

	Creation of Modules
		Modules over dedekind domains can be created from orders of number fields and function fields and combinations of ideals and vector space elements.
		Create the free module O^n where O is a dedekind domain.
		s with entries in a dedekind domain or field of fractions of a dedekind domain.
	cso.ulb.ac.be /~dleemans/magma/text803.htm (526 words)

	11 Definitions for a Dedekind Domain
		Thus r[x] is a fractional ideal, and inside a dedekind domain, every fractional ideal is invertable.
		Let h be a p primary ideal in a dedekind domain r.
		All ideals are invertable, and by condition 5, r/s is dedekind.
	www.mathreference.com /id-dd,defs.html (675 words)

	HJM, Vol. 25, No. 1, 1999
		A saturated multiplicative subset S of R different from the unit group of R is a GCD-set if gcd(a,b) exists for each element a and b of S. We study the structure of GCD-sets of R, with emphasis on the case where R is a Dedekind domain.
		We show that if R is atomic, then each GCD-set is generated by completely irreducible elements, and that if R is a Dedekind domain and x is a nonzero nonunit of R, then some positive power of x has a completely irreducible factor.
		Let R be a Dedekind domain with torsion realizable pair {Cl(R),A}.
	www.math.uh.edu /~hjm/Vol25-1.html (1168 words)

	Homomorphisms between Modules (Site not responding. Last check: 2007-10-23)
		It is possible to create a homomorphism between two modules, take the image and kernel of such and verify that these are submodules of the codomain and domain respectively.
		The Hom--module can also be created as a module of a dedekind domain.
		The module is over the same dedekind domain as M and N. IsSubmodule(M, N) : ModDed, ModDed -> BoolElt, Map
	modular.fas.harvard.edu /docs/magma/htmlhelp/text809.htm (350 words)

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