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Topic: Integral domain

Related Topics

Prime element

Glossary of ring theory

Prime ideal

Ring ideal

Ring (mathematics)

Principal ideal

Principal ideal domain

Commutative ring

Euclidean domain

Gaussian integer

Integral calculus

Quasigroup

Lebesgue integral

Prime

Algebra over a field

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	Integral domain - Wikipedia, the free encyclopedia
		Integral domains are generalizations of the integers and provide a natural setting for studying divisibility.
		Alternatively and equivalently, integral domains may be defined as commutative rings in which the zero ideal {0} is prime, or as the subrings of fields.
		Integral domains are also characterized by the condition that they are reduced and irreducible.
	en.wikipedia.org /wiki/Integral_domain (907 words)

	PlanetMath: integral domain
		It is also not very closely related to the notion of integral, which is applied to ring elements, or that of integral closure, which is applied to extensions of rings, although these concepts are normally applied to integral domains.
		Commutative integral domains have fraction fields, which play the role of the rational numbers, and they each have a characteristic (which is either a prime number or zero).
		This is version 10 of integral domain, born on 2001-10-19, modified 2004-04-24.
	planetmath.org /encyclopedia/IntegralDomain.html (202 words)

	PlanetMath: an artinian integral domain is a field
		Therefore, every artinian integral domain is a field.
		"an artinian integral domain is a field" is owned by saforres.
		This is version 3 of an artinian integral domain is a field, born on 2002-07-02, modified 2002-07-02.
	planetmath.org /encyclopedia/AnArtinianIntegralDomainIsAField.html (79 words)

	Principal ideal domain - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-06)
		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).
		Examples are the ring of integers, all fields, and rings of polynomials in one variable with coefficients in a field.
		All Euclidean domains are principal ideal domains, but the converse is not true.
	en.wikipedia.org /wiki/Principal_ideal_domain (316 words)

	Euclidean domain - Wikipedia, the free encyclopedia
		In abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm can be used.
		In fact, if I is a nonzero ideal of a Euclidean domain D and a nonzero a in I is chosen to minimize v(a) over all elements of I, then I = aD.
		The name Euclidean domain comes from the fact that the extended Euclidean algorithm can be carried out in any Euclidean domain.
	en.wikipedia.org /wiki/Euclidean_domain (458 words)

	Abstract algebra:Integral domains - Wikibooks, collection of open-content textbooks
		Integral domains are a useful tool for studying the conditions under which concepts like divisibility and unique factorization are well-behaved.
		For this reason an integral domain is sometimes called a cancellation ring.
		Theorem: Let R be an integral domain and let R[x] be the ring of polynomials in powers of x whose coefficients are elements of R.
	en.wikibooks.org /wiki/Abstract_algebra:Integral_domains (491 words)

	Reference.com/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.
		An alternative characterization of Dedekind domains is that an integral domain R is a Dedekind domain if and only if the localization of R at each prime ideal P of R is a discrete valuation ring.
		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).
	www.reference.com /browse/wiki/Dedekind_domain (493 words)

	APPENDIX J
		The ordering of a domain D implies the cancellation law, and therefore that D is an integral domain.
		The countably infinite domain of all integers J, is an integral domain.
		Similarly the Gaussian integers J[i] is an integral domain.
	graham.main.nc.us /~bhammel/FCCR/apdxJ.html (6145 words)

	Integral Domain (Site not responding. Last check: 2007-10-06)
		A domain is a ring with left and right cancellation.
		Given a nonzero element c, c*x is a permutation on the ring by cancellation.
		Since c was arbitrary, all elements are left invertable, and by an earlier result, the ring is a division ring.
	www.mathreference.com /ring,intd.html (224 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.
		Let D be an integral domain with quotient field F, and let I be an ideal of D that is invertible when considered as a fractional ideal.
		Let D be an integral domain with quotient field Q, and let F be a finite extension field of Q. If D* is the set of all elements of F that are integral over D, then D* is a Dedekind domain.
	www.math.niu.edu /~beachy/aaol/commutative.html (2296 words)

	Integral Institute: Integral Training (Site not responding. Last check: 2007-10-06)
		Integral Commons has everything you need to understand and participate in any of the 30 domains of integral practice.
		That is the first tier of integral learning—namely, all of the partners of a particular domain will form a mega-feedback loop on how to apply the Integral Approach in their domains; which types of approaches work, which don’t; advice and practical tips on how to proceed; trial and error in the application of Integralism.
		However, not only will each domain be pioneering ways to apply integral approaches in their own fields—personal to professional—each of the domains will be talking with each other.
	www.integralinstitute.org /multi.htm (1031 words)

	Prof. Weile's Research
		Frequency domain CEM methods are more efficient for the simulation of narrowband phenomena and phenomena involving loss, but they can not handle nonlinear problems and are very inefficient for the simulation of broadband phenomena or moving objects.
		Time domain methods have the exact opposite properties; they are efficient over broad bands, can simulate nonlinear and moving phenomena, but the simulation of loss is difficult.
		Moreover, if the object under study is homogeneous, the unknown current in the integral equation exists only on the surface, as opposed to filling the whole volume of the structure.
	www.ece.udel.edu /~weile/research.html (827 words)

	[No title] (Site not responding. Last check: 2007-10-06)
		If q is not of the domain type Dom::Fraction(R), it is considered as a rational expression which is converted into a fraction over R and multiplied with q.
		It uses the method "random" of the domain R to randomly generate its numerator and denominator.
		This method tries to convert r into an element of the domain T, or, if T is not a domain, to the domain type of T.
	www.sciface.com /STATIC/DOC30/eng/Dom_Fraction.html (1350 words)

	PlanetMath: Euclidean domain
		Any Euclidean domain is also a principal ideal domain and therefore also a unique factorization domain.
		But even more important, on Euclidean domains we can define gcd and use Euclid's algorithm.
		This is version 6 of Euclidean domain, born on 2002-05-27, modified 2002-07-28.
	planetmath.org /encyclopedia/EuclideanRing.html (110 words)

	Weakly integrally closed domains:\\minimum polynomials of matrices
		If D is a subring of an integral domain R, we say that R is a separable quadratic extension of D if R is a subring of a separable quadratic extension field of the quotient field K of D.
		Of course R is integrally closed, being a unique factorization domain, but the ring R[2i] is not, and R[2i] is weakly integrally closed because it is a free quadratic extension of the Krull domain R.
		Characterize the weakly integrally closed algebras k[M] where M is a submonoid of the free monoid on one generator, and k is a field.
	www.math.fau.edu /Richman/Docs/weakly.html (3294 words)

	Field of fractions - Psychology Central (Site not responding. Last check: 2007-10-06)
		In mathematics, every integral domain can be embedded in a field; the smallest field which can be used is the field of fractions of the integral domain.
		The elements of the field of fractions of the integral domain R have the form a/b with a and b in R and b ≠ 0.
		The functor from C to the category of fields which takes every integral domain to its fraction field and every homomorphism to the induced map on fields (which exists by the universal property) is the left adjoint of the forgetful functor from the category of fields to C.
	psychcentral.com /psypsych/Quotient_field (398 words)

	Dedekind Domains (Site not responding. Last check: 2007-10-06)
		However, it is not a Dedekind domain because it is not an integral domain.
		is integrally closed, and by Proposition 5.2.5 it is Noetherian.
		Thus unique factorization holds for ideals in a Dedekind domain, and it is this unique factorization that initially motivated the introduction of rings of integers of number fields over a century ago.
	modular.fas.harvard.edu /papers/ant/html/node17.html (819 words)

	Modern Algebra II Lecture Notes, 09/10/03 (Site not responding. Last check: 2007-10-06)
		Definition: An integral domain D is a commutative ring with unity 1 (not equal to 0) that has no divisors of 0.
		Proof: By definition, every integral domain is a commutative ring with unity (not equal to 0).
		Since D is an integral domain and we've assumed that x-y is non-zero (since x and y are distinct), we must have that a equals zero, contradiction.
	www.assumption.edu /Alfano/MAT352-FA03/Notes/091003.html (355 words)

	Math 410
		A companion type of question could be: are there domains in which the concepts are equivalent.
		Here's an example of an Integral Domain which is not a PID, yet irreducible elements are also prime elements.
		However, some of the early pit falls in the proof of Fermat's Last Theorem, centered about the fact that the integral domains that were constructed to demonstrate the truth of the theorem were not UFDs.
	mathserv.monmouth.edu /coursenotes/kuntz/math411/m41101.htm (918 words)

	Matrix Reference Manual: Algeraic Structures
		The characteristic of an integral domain is the minimum number of 1's that must be added together to equal 0.
		The characteristic of an integral domain or field is always either 0 or a prime number.
		F (also called a rational domain) is an integral domain in which every non-zero element has a multiplicative inverse.
	www.ee.uwa.edu.au /~roberto/teach/matrix/vector.html (880 words)

	Two-dimensional time-domain volume integral equations for scattering of inhomogeneous objects
		This paper proposes a time-domain volume integral equation based method for analyzing the transient scattering from a two-dimensional inhomogeneous cylinder by invoking the volume equivalence principle for both the transverse magnetic and electric cases.
		For the transverse electric case, because of the electric charges induced inside and on the surface of the cylinder, the electric flux is defined on the edges of the triangles, and expanded in space in terms of two-dimensional surface roof-top basis functions.
		The time-domain volume integral equation is solved by using a marching-on-in-time scheme.
	www.agu.org /pubs/crossref/2003/2000RS002605.shtml (248 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)

	Ring Theory
		Give examples of a noncommutative ring with zero divisors, a noncommutative division ring, and integral domain, a UFD, a PID, a Euclidean domain and examples which show that ID Be sure to justify that your examples have or do not have the requisite properties.
		Prove that a Euclidean integral domain is a PID.
		This is the converse of a well-known theorem.
	www.math.dartmouth.edu /graduate-students/syllabi/sample-questions/algebra/node3.html (274 words)

	7.9.2 Contour integral evaluation
		Using the divergence theorem, the contour integral can be expanded into an area integral in two dimensions or a volume integral in three dimensions, over a finite domain surrounding the crack front.
		Contributions to the contour integral due to concentrated loads in the domain are not included; instead, the mesh must be modified to include a small element and a distributed load must be applied to this element.
		The contour integral evaluation capability in ABAQUS/Standard assumes that the elements that lie within the domain used for the calculations are quadrilaterals in two-dimensional or shell models or bricks in continuum three-dimensional models.
	www.rpi.edu /AFS/home/85/millem/public_html/v6.4/books/usb/pt03ch07s09aus47.html (4549 words)

	Dom::Fraction -- the field of fractions of an integral domain (Site not responding. Last check: 2007-10-06)
		-- the field of fractions of an integral domain
		The result is a factored object, i.e., an element of the domain type
		is not a domain, to the domain type of
	www.sciface.com /STATIC/DOC25/de/Dom/Fraction.shtml (841 words)

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