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Topic: Principal ideal

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	Traces, ideals, and arithmetic means -- Kaftal and Weiss 99 (11): 7356 -- Proceedings of the National Academy of ...
		ideal I are defined in ref. 4 as I
		ideals in the literature are am-closed, i.e., I = I
		(I) of an am-closed ideal I is hereditary
	www.pnas.org /cgi/content/full/99/11/7356 (2657 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.
		It is not principal, since for example the ideal generated by 2 and X cannot be generated by a single polynomial.
	en.wikipedia.org /wiki/Principal_ideal_domain (306 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.
		In a principal ideal domain, any two elements have a greatest common divisor (and may have more than one).
		Every principal ideal domain is Noetherian and a unique factorization domain.
	www.ebroadcast.com.au /lookup/encyclopedia/pr/Principal_ideal_domain.html (138 words)

	Principal ideal domain -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-16)
		In a principal ideal domain, any two elements have a (The largest integer that divides without remainder into a set of integers) greatest common divisor, and almost always have more than one.
		Every principal ideal domain is a (Click link for more info and facts about unique factorization domain) 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 (Click link for more info and facts about principal ideal domain) principal ideal domain that is not a (Click link for more info and facts about euclidean domain) euclidean domain is the ring (Wilson, J. "A Principal Ring that is Not a Euclidean Ring." Math.
	www.absoluteastronomy.com /encyclopedia/p/pr/principal_ideal_domain.htm (342 words)

	Ideal Science: Bulletin Board Software
		Ideal Science, Inc. collects this information as well, but does so exclusively for the following uses: (1) anonymous statistical purposes; or (2) administration of our web site and servers, and to improve our services.
		Ideal Science, Inc. collects data in aggregate form and data is not recorded or stored for individual visitors.
		Ideal Science, Inc. uses cookies to track such as the time/date of the visit, the time/date of last visit, the page viewed, the referrer, transaction information for eCommerce pages, and other data.
	www.idealscience.com /site/about/privacy.aspx (472 words)

	Ring ideal : Ideal (rings) (Site not responding. Last check: 2007-10-16)
		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 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 "ideal number": ideals were seen as a generalization of the concept of number.
	www.city-search.org /id/ideal-(rings).html (1630 words)

	Encyclopedia: Principal ideal domain
		In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers.
		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. More specifically: a left principal ideal of R is a subset of R of the form Ra := {ra : r in R...
		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.
	www.nationmaster.com /encyclopedia/Principal-ideal-domain (747 words)

	PlanetMath: principal ideal ring
		generated by a single ring element, is called a principal ideal ring.
		is also an integral domain, it can be called a principal ideal domain.
		This is version 2 of principal ideal ring, born on 2004-08-23, modified 2004-08-23.
	planetmath.org /encyclopedia/PrincipalIdealRing.html (84 words)

	Principal ideal (Site not responding. Last check: 2007-10-16)
		In Ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a singleelement 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.
		In principal ideal domains, this allows us to calculate greatest common divisors of elementsof the ring, up to multiplication by a unit ; we definegcd(a,b) to be any generator of the ideal (a,b).
	www.therfcc.org /RFCC/principal-ideal-210826.html (489 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 (103 words)

	Applicability and Efficiency of Near-Optimal Spatial Encoding for (Site not responding. Last check: 2007-10-16)
		An analysis of the distribution of principal angles is introduced and applied in several example cases to quantitatively describe the suitability of a basis set derived from a specific image estimate for the spatial encoding of a given field-of-view.
		The k'th principal angle represents the minimum angle existing between a vector in one subspace (a principal vector in the first subspace) and a vector in the other subspace (a principal vector in the second subspace) with the restriction that neither of these two vectors contributed to the formation of the first (k-1)'th principal angles.
		Ideal encoding, the limiting case, occurs only when all these principal angles are zero, and the number of encoding vectors equal to the matrix rank are used.
	splweb.bwh.harvard.edu:8000 /pages/papers/zientara/svd/svd_update.html (6833 words)

	Encyclopedia: Principal ideal
		In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers.
		Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields.
		In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring.
	www.nationmaster.com /encyclopedia/Principal-ideal (1459 words)

	National Association of Secondary School Principals. NASSP Bulletin: Portrait of the "ideal principal": ... (Site not responding. Last check: 2007-10-16)
		This single and idealized view of the principal's role in a reforming school ignores both the complexity and contextual nature of school leadership.
		For example, Hallinger and Murphy (1985) found that principals in communities with a lower socioeconomic status tended to be both controlling and coordinating in their administrative styles, whereas principals in communities with a high socioeconomic status relied on more coordination.
		Because principals are hired and evaluated by school boards and superintendents and because schools are dependent on districts for many of their resources (Peterson 1984; Peterson, Murphy, and Hallinger 1987), principals are influenced by their district superiors' concept of a principal's role.
	www.findarticles.com /p/articles/mi_qa3696/is_200009/ai_n8904486 (1384 words)

	Ideals and Quotients
		Given an integral ideal I of O, return two elements of the field of fractions of O that form a two-element normal presentation for I, as well as an integer g such that I is g-normal.
		The denominator of the fractional ideal I. This is the smallest positive integer d such that dI is an integral ideal.
		Returns the basis matrix for the ideal I of O. The basis matrix consists of the elements of a Z-basis for the ideal written as rows of rational coefficients with respect to the power basis of the number field K of which O is an order.
	www.math.wisc.edu /help/magma/text455.html (1187 words)

	principal ideal
		An ideal of a ring R is called principal if there is an element a of R such that.
		principal ideal domain In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is.
		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 …
	www.marymags.net /principal-ideal.html (452 words)

	MTH-2A24 : Algebra II
		Apart from the general notions of rings, ideals and homomorphisms, the course deals with the theory of polynomials, an introduction to field theory and the theory of factorization.
		The notions of principal ideal and divisibility are steps to higher arithmetics with divisibility theory, and to the theory of polynomial rings.
		The next portion of abstract theory is the theory of principal ideal domains and the theory of Euclidean domains.
	www.mth.uea.ac.uk /maths/syllabuses/0001/2A2401.html (499 words)

	[No title]
		Note that principal ideal rings lie between principal ideal domains and Noetherian rings.
		We use the fact that a ring is a principal ideal ring if and only if it is isomorphic to a direct product of principal ideal domains and Artinian chain rings.
		Using Theorem 4 these results can be generalised to principal ideal rings and the restriction on the length of the code can be removed.
	www-calfor.lip6.fr /ICPSS/papers/40Sa/40Sa.htm (959 words)

	PlanetMath: PID (Site not responding. Last check: 2007-10-16)
		is an integral domain where every ideal is a principal ideal.
		See Also: UFD, irreducible, ideal, integral domain, Euclidean domain, Euclidean valuation, proof that an Euclidean domain is a PID, motivation for Euclidean domains
		Cross-references: UFD, prime, irreducible, maximal, principal ideal, ideal, integral domain
	planetmath.org /encyclopedia/PID.html (119 words)

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