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	Noetherian ring - Wikipedia, the free encyclopedia
		In abstract algebra, a Noetherian ring is a ring that satisfies the ascending chain condition on ideals.
		The Noetherian property is central in ring theory and in areas that make heavy use of rings, such as algebraic geometry.
		This early result was the first to suggest that Noetherian rings possessed a deep theory of dimension.
	en.wikipedia.org /wiki/Noetherian_ring (474 words)

	Noetherian - Wikipedia, the free encyclopedia
		In mathematics, Noetherian is an adjective derived from the name of Emmy Noether, describing objects that satisfy an ascending chain condition on certain kinds of subobjects.
		A Noetherian ring is a ring that satisfies the ascending chain condition on ideals.
		A Noetherian module is a module that satisfies the ascending chain condition on submodules.
	www.wikipedia.org /wiki/Noetherian (134 words)

	PlanetMath: Let N be a submodule of M. M is a Noetherian module iff M/N and N are Noetherian modules. (Site not responding. Last check: 2007-11-07)
		PlanetMath: Let N be a submodule of M. M is a Noetherian module iff M/N and N are Noetherian modules.
		Let N be a submodule of M. M is a Noetherian module iff M/N and N are Noetherian modules.
		This is version 3 of Let N be a submodule of M. M is a Noetherian module iff M/N and N are Noetherian modules.
	planetmath.org /encyclopedia/FactorModule.html (116 words)

	Station Information - Noetherian ring
		In mathematics, a ring is called Noetherian if, intuitively speaking, it is not "too large" as expressed by a certain finiteness condition on its idealss.
		Noetherian rings are named after the mathematician Emmy Noether, who developed much of their theory.
		An example of a ring that's not Noetherian is a ring of polynomials in infinitely many variables: the ideal generated by these variables cannot be finitely generated.
	www.stationinformation.com /encyclopedia/n/no/noetherian_ring.html (290 words)

	PlanetMath: example of a right noetherian ring that is not left noetherian (Site not responding. Last check: 2007-11-07)
		The approach given here uses the fact that a ring is right noetherian if all of its right ideals are finitely generated.
		"example of a right noetherian ring that is not left noetherian" is owned by smw.
		This is version 13 of example of a right noetherian ring that is not left noetherian, born on 2004-03-19, modified 2005-10-01.
	planetmath.org /encyclopedia/ExampleOfRightNoetherianRingThatIsNotLeftNoetherian.html (276 words)

	Noetherian (Site not responding. Last check: 2007-11-07)
		In mathematics, Noetherian is an adjective that describes objects that satisfy an ascending chain condition on certain kinds of subobjects.
		A ring is a Noetherian ring if it satisfies the ascending chain condition on ideals.
		A module is a Noetherian module if it satisfies the ascending chain condition on submodules.
	pedia.newsfilter.co.uk /wikipedia/n/no/noetherian.html (124 words)

	[No title]
		The basic idea is to start with a typical Noetherian integral domain $R$ such as a polynomial ring in several indeterminates over a field and to look for unusual Noetherian and non-Noetherian extension rings inside a homomorphic image $S$ of an ideal-adic completion $R^*$ of $R$.
		Then $A$ is Noetherian (in fact a 2-dimensional regular local domain \footnote{ This example constructed by Nagata (historically) is the first occurence of a 2-dimensional regular local domain containing a field of characteristic zero that fails to be pseudo-geometric.
		The Noetherian property of $B$ is implied by the flatness property of the map $B_0 \to D_0[1/a]$.
	www.math.purdue.edu /~heinzer/preprints/bou16.tex (5085 words)

	PlanetMath: noetherian (Site not responding. Last check: 2007-11-07)
		is left noetherian if it is noetherian as a left module over itself (i.e.
		is a noetherian module), and right noetherian if it is noetherian as a right module over itself (i.e.
		This is version 2 of noetherian, born on 2002-02-24, modified 2003-09-20.
	planetmath.org /encyclopedia/Noetherian2.html (195 words)

	Encyclopedia: Noetherian module (Site not responding. Last check: 2007-11-07)
		In ring theory, if R is a ring and M is a module over R, then M is Noetherian if M satisfies the ascending chain condition on its submodules when they are ordered by inclusion.
		Noetherian modules are named in honor of Emmy Noether.
		Any finitely generated module over a Noetherian ring is a Noetherian module.
	www.nationmaster.com /encyclopedia/Noetherian-module (110 words)

	UWM Math: Noetherian Rings (Site not responding. Last check: 2007-11-07)
		The concepts of noetherian and artinian rings were abstracted from specific commutative examples in the 1920's.
		Natural examples of non-commutative rings need not be noetherian; nevertheless, the noetherian hypothesis is very useful and fortunately does hold in many cases.
		While many interesting ring theoretic results were proven in between, it is probably fair to say that the modern study of non-commutative noetherian rings began with A. Goldie's work in 1958-1960 giving necessary and sufficient conditions for a ring to have a semisimple ring of fractions.
	www.uwm.edu /Dept/Math/Research/Algebra/noetherian/noetherian.html (467 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		By the induction hypothesis $R/J^{n-1}$ is artinian iff it is noetherian.
		Thus, as claimed, $_RR$ is artinian iff it is noetherian.
		For a left noetherian ring, on the other hand, the radical $J(R)$ can be far from nil, and yet as we shall now see, every nil one-sided ideal is nilpotent.
	darkwing.uoregon.edu /~anderson/math649/lecture37.html (1009 words)

	Noetherian Girl (Site not responding. Last check: 2007-11-07)
		And that's cos we are living in a Noetherian world, And I am a Noetherian girl.
		You know that we are living in a Noetherian world, And I am a Noetherian girl.
		A Noetherian, a Noetherian, a Noetherian, a Noetherian world.
	math.colorado.edu /~rmg/weird/noeth.html (237 words)

	Noetherian Centralizing Hopf Algebra Extensions And Finite Morphisms Of Quantum Groups (ResearchIndex)
		Noetherian Centralizing Hopf Algebra Extensions And Finite Morphisms Of Quantum Groups (ResearchIndex)
		Noetherian Centralizing Hopf Algebra Extensions And Finite Morphisms Of Quantum Groups (1998)
		We study finite centralizing extensions A ae H of Noetherian Hopf algebras.
	citeseer.ist.psu.edu /letzter98noetherian.html (367 words)

	[No title]
		Let $R$ be a Noetherian integral domain with field of fractions $K$ and let $a$ be a nonzero nonunit of $R$.
		In \cite{HRW3} we consider in the case where $R$ is a semilocal Noetherian integral domain and $a$ is an element of the Jacobson radical of $R$ the condition that the embedding $U_0 \to R^*[1/a]$ is flat.
		It looks to me that to show this comes down to showing that a directed union of Noetherian domains of a certain type have the property that their union is Noetherian.
	www.math.purdue.edu /~heinzer/preprints/eno11.tex (3742 words)

	Research in Algebra \| Ring Theory
		Later, it was realised that commutative noetherian rings are one of the building blocks of modern algebraic geometry, leading to their study both abstractly and in examples.
		As in the commutative case, non-commutative noetherian rings are studied in abstraction and in examples.
		Classic examples of noetherian rings include the co-ordinate rings of affine varieties, rings of differential operators on smooth algebraic varieties, universal enveloping algebras of finite dimensional Lie algebras and group algebras of polycyclic-by-finite groups.
	www.maths.gla.ac.uk /research/groups/algebra/rings.htm (986 words)

	ABSTRACT ALGEBRA ON LINE: Ideal Theory of Commutative Rings (Site not responding. Last check: 2007-11-07)
		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.
		One important consequence of the generalized principal ideal theorem is that any Noetherian ring satisfies the descending chain condition for prime ideals.
		If R is Noetherian, and has a chain of prime ideals of length n, but none longer, then we say that R has Krull dimension equal to n.
	www.math.niu.edu /~beachy/aaol/commutative.html (2296 words)

	The Noetherian Ring at Berkeley
		Emmy Noether, the Noetherian Ring is an organization of graduate students, postdocs, and professors in the
		As a group of predominantly female mathematicians, the Noetherian Ring at Berkeley is far from unique.
		Noetherian Ring at the University of Wisconsin (Madison),
	math.berkeley.edu /~nring (413 words)

	Citations: An introduction to noncommutative noetherian rings - Goodearl, Warfield (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		....left noetherian ring and suppose R to be prime, i.e.
		noetherian pi rings) artinian rings, principal ideal rings, hereditary noetherian prime (HNP) rings with enough invertible ideals,.
		i) If R 1 and R 2 are prime noetherian rings, then a nonzero R 1 R 2 bimodule that is finitely generated on each side and torsion free (see, e.g.
	sherry.ifi.unizh.ch /context/61581/0 (2069 words)

	Chains of Modules (Site not responding. Last check: 2007-11-07)
		Another synonym for noetherian is "ascending chain condition", or ACC.
		If r is a ring, r is a left r module, and the submodules are the ideals.
		Thus a noetherian ring has no infinite ascending chains of ideals, and an artinian ring has no infinite descending chains of ideals.
	www.mathreference.com /mod-acc,intro.html (236 words)

	Atlas: Regularity on varieties over non-noetherian valuation rings by Hagen Knaf (Site not responding. Last check: 2007-11-07)
		It is therefore natural to ask for a generalization of this notion to non-noetherian schemes.
		In the sequel coherent regular rings were investigated, showing the similiarity of this class of rings with the class of noetherian regular rings as well as significant differences.
		In the talk the local rings of points on an integral separated scheme of finite type over a Pruefer domain R - an R-variety for short - are considered: Let O be the local ring in a point x of such a scheme and let M be its maximal ideal.
	atlas-conferences.com /cgi-bin/abstract/cacv-36 (341 words)

	The ascending tree condition
		From a classical point of view, every finitely generated module over a Noetherian ring is finitely presented-in particular, every Noetherian ring is coherent-so this also provides an adequate classical theory.
		For example, to prove that a quotient of a Noetherian module is Noetherian, you lift a chain of finitely generated ideals from the quotient to the module.
		Define a Noetherian module to be a module that satisfies the ATC on finitely generated submodules.
	www.math.fau.edu /richman/docs/new-acc.htm (2439 words)

	AMCA: Prime Properties of Irreducible Elements in a Birational Extension of A Noetherian UFD by Aihua Li (Site not responding. Last check: 2007-11-07)
		AMCA: Prime Properties of Irreducible Elements in a Birational Extension of A Noetherian UFD by Aihua Li Atlas Mathematical Conference Abstracts
		Let B be a finitely generated birational extension of R; that is, B is a Noetherian integral domain between R and the quotient field of R. A finitely generated birational extension of a Noetherian UFD R has the form: R[g
		It is well known that a Noetherian integral domain is a UFD if and only if every irreducible element of it generates a (principal) prime ideal.
	at.yorku.ca /c/a/b/w/23.htm (200 words)

	Noetherian ring : Noetherian (Site not responding. Last check: 2007-11-07)
		In mathematics, a ring is called Noetherian if, intuitively speaking, its ideals are not "too large", expressed by a certain finiteness condition.
		Every field is trivially Noetherian, since a field F has only two ideals - F and {0}.
		All is still licensed under the GNU FDL.
	www.termsdefined.net /no/noetherian.html (507 words)

	Ivan Gotchev (Site not responding. Last check: 2007-11-07)
		Let X be a Noetherian topological space and P be the set of all closed irreducible subsets of X, ordered by inclusion.
		There exists an example of a countable Noetherian topological space Y such that h(Y)=2 and which is not sequential.
		Let X be a Noetherian topological space in which every irreducible subset F has common (general) point x, i.e.
	www.utm.edu /~jschomme/topology/c/a/a/h/24.htm (452 words)

	Le résultat de votre recherche (Site not responding. Last check: 2007-11-07)
		If $R$ is a filtered ring such that the Rees ring $\widetilde{R}$ of $R$ is left Noetherian, and $T$ is a left Ore set of the associated graded ring $G(R)$ consisting of homogeneous elements, then the saturated set $S=\{s\in R,\ \sigma(s)\in T\}$ is a left Ore set of $R$.
		If $R$ is a ring graded by the integers, which is noetherian and left graded regular, such that every finitely generated graded projective $R$-module is graded stably free, then it is shown that the corresponding ungraded property holds.
		Zariski rings with noetherian Rees ring and with commutative associated graded ring are investigated.
	www.math.jussieu.fr /~keller/semalg/li.html (2114 words)

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