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Topic: Ascending chain condition

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Emmy Noether

	Chain
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		Hawaiian-Emperor seamount chain The Hawaiian-Emperor seamount chain is composed of the 1963, geologist Tuzo Wilson hypot...
		Simpson Chain The Simpson Chain or Simpson Lever Chain was an 1895.
	www.brainyencyclopedia.com /topics/chain.html (1488 words)

	Ascending chain condition - Wikipedia, the free encyclopedia
		The ascending chain condition on P is equivalent to the maximum condition: every nonempty subset of P has a maximal element.
		Similarly, the descending chain condition is equivalent to the minimum condition: every nonempty subset of P has a minimal element.
		A totally ordered set that satisfies the descending chain condition is called a well-ordered set.
	en.wikipedia.org /wiki/Ascending_chain_condition (166 words)

	Noetherian ring - Wikipedia, the free encyclopedia
		In abstract algebra, a Noetherian ring is a ring that satisfies the ascending chain condition on ideals.
		Rings of polynomials over fields have many special properties; properties that follow from the fact that polynomial rings are not, in some sense, "too large".
		Emmy Noether first discovered that the key property of polynomial rings is the ascending chain condition on ideals.
	en.wikipedia.org /wiki/Noetherian_ring (474 words)

	PlanetMath: maximal condition
		A group satifies the maximal condition if and only if the group and all its subgroups are finitely generated.
		Similar properties are useful in other classes of algebraic structures: see for example the Noetherian condition for rings and modules.
		This is version 2 of maximal condition, born on 2003-10-04, modified 2003-10-04.
	planetmath.org /encyclopedia/MaximalCondition.html (113 words)

	chain
		A chain is a reliable machine component, which transmits power by means of tensile forces, and is used primarily for power transmission and conveyance systems.
		A chain is a measurement of length equivalent to 22 yards (20.12 metres), which is one tenth of a furlong or one eightieth of a mile.
		The distance of 22 yards is the length of the pitch (from wicket to wicket) in cricket.
	www.fact-library.com /chain.html (192 words)

	Chains of Modules (Site not responding. Last check: 2007-10-20)
		If you like set theory, think of a chain as a map from the ordinals into the submodules of m, where the submodules increase or decrease as the ordinals advance.
		Noether (biography) investigated modules with no infinite ascending chains; and these modules are now called noetherian.
		Another synonym for noetherian is "ascending chain condition", or ACC.
	www.mathreference.com /mod-acc,intro.html (236 words)

	Science Fair Projects - Noetherian
		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.
		Emmy Noether was the first to study the ascending and descending chain conditions for rings.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Noetherian (245 words)

	Noetherian (Site not responding. Last check: 2007-10-20)
		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.
		The concept is named for Emmy Noether, who first studied the ascending and descending chain conditions for rings.
	pedia.newsfilter.co.uk /wikipedia/n/no/noetherian.html (124 words)

	Emmy Noether - Wikipedia, the free encyclopedia
		The results of Noether's theorem are part of the fundamentals of modern physics, which is substantially based on the properties of symmetries.
		In 1921, Noether introduced the ascending chain condition for ideals in a commutative ring, and proved the existence of primary decompositions for such rings (a result known as the Lasker-Noether theorem).
		Rings satisfying the ascending chain condition on ideals are now known as Noetherian rings.
	en.wikipedia.org /wiki/Emmy_Noether (499 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		By the descending chain condition applied in the case where the $n_i=1$ we see that there are only finitely many maximal ideals in an Artinian ring.
		Continuing this process, we see that, the descending chain condition for $A$ implies that there must exist some multiple $y$ of $x_0$ with the property that $\Ann(Ay)$ is a maximal ideal.
		The descending chain condition for ideals thus implies that $J^r$ must be the zero ideal for some $r$.
	www.imsc.res.in /~kapil/geometry/caag/finite.tex (2783 words)

	Finite Rings (Site not responding. Last check: 2007-10-20)
		It is clear that such a ring satisfies the descending chain condition for ideals.
		The ring also satisfies the ascending chain condition; any ascending chain of ideals is constant after a finite stage.
		The given condition on f can be restated by saying that H(f) is a unit in the ring of formal power series and L(f) is in M[X] where M is the maximal ideal of the ring A.
	www.imsc.ernet.in /~kapil/geometry/caag/finite.html (2519 words)

	Noetherian ring
		In mathematics, a ring is called Noetherian if, intuitively speaking, its ideals are not "too large", expressed by a certain finiteness condition.
		This can be rephrased as "the poset of (two-sided) ideals in R under inclusion has the ascending chain condition".
		The ring R is called right-Noetherian if the above conditions are true for right ideals, and it is called Noetherian if it is both left-Noetherian and right-Noetherian.
	www.ebroadcast.com.au /lookup/encyclopedia/no/Noetherian.html (264 words)

	philosophy notes
		Since the descending chain condition treats all left ideals the hypothesis of this theorem is phrased in second order rather than first order logic.
		The fact that a polynomial ring over a field is Noetherian yields to a descending chain condition on closed subgroups.
		Both proofs use the chain conditions and induction on rank as a substitute for cardinality.
	www.math.uic.edu /~jbaldwin/pub/phil.html (2052 words)

	PlanetMath: ascending chain condition
		when ordered by inclusion) satisfies the ascending chain condition or ACC if there does not exist an infinite ascending
		Cross-references: descending chain condition, chain, infinite, satisfies, inclusion, subsets, collection, partially ordered set
		This is version 4 of ascending chain condition, born on 2001-11-23, modified 2004-05-01.
	planetmath.org /encyclopedia/AscendingChainCondition.html (72 words)

	Graduate Seminar
		One property people often ask as a first question about a ring is what are some basic properties of its ideals.
		One property that turns out to be quite useful in proving many basic results about a ring is contained in information concerning chains of ideals.
		Two seemingly dueling notions are the ascending chain condition (ACC) and the descending chain condition (DCC).
	www.math.uga.edu /~clint/vigre/semSu03.htm (662 words)

	Commutative Algebra at the University of Zimbabwe (Site not responding. Last check: 2007-10-20)
		The ascending chain condition and descending chain condition in rings.
		The relationship between maximal/minimal conditions and nilpotency in rings.
		The right/left annihilator of a subset of a ring.
	www.uz.ac.zw /science/maths/courses/hmth035.htm (118 words)

	Ascending Chain Condition Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-10-20)
		Looking For ascending chain condition - Find ascending chain condition and more at Lycos Search.
		Find ascending chain condition - Your relevant result is a click away!
		Look for ascending chain condition - Find ascending chain condition at one of the best sites the Internet has to offer!
	www.karr.net /search/encyclopedia/Ascending_chain_condition (364 words)

	PlanetMath: descending chain condition
		, ordered by inclusion) satisfies the descending chain condition or DCC if there does not exist an infinite descending
		Cross-references: ascending chain condition, chain, infinite, satisfies, inclusion, subsets, collection, partially ordered set
		This is version 3 of descending chain condition, born on 2001-11-23, modified 2004-05-01.
	planetmath.org /encyclopedia/DescendingChainCondition.html (72 words)

	Dilworth, Robert P. (1939-01-01) The structure and arithmetical theory of non-commutative residuated lattices. ...
		It is shown that under certain general conditions each of these operations may be defined in terms of the other.
		It is shown that each element of a modular lattice in which suitable chain conditions hold, may be represented as a product of irreducibles; and if there are two such decompositions, the number of irreducibles is the same and they are similar in pairs.
		In the first part the structure of ideal lattices in the vicinity of the unit element is characterized in terms of arithmetical and semi-arithmetical lattices.
	resolver.caltech.edu /CaltechETD:etd-05222003-111959 (429 words)

	Ascending chain condition (Site not responding. Last check: 2007-10-20)
		In mathematics, a poset P is said to satisfy the ascending chain condition (ACC)if every ascending chain a1 ≤ a2 ≤.
		All is still licensed under the GNU FDL.
		B may multiply asexually again; in the simpler cases, however, it.
	www.termsdefined.net /as/ascending-chain-condition.html (210 words)

	Atlas: Chain Conditions on Quotient Finite (Uniform) Dimensional Modules by Toma Albu (Site not responding. Last check: 2007-10-20)
		This paper is motivated by the recent work of C. Faith, Quotient finite dimensional modules with ACC on subdirectly irreducible submodules are Noetherian (to appear in Comm.
		Algebra, 1999), who has proved that a quotient finite dimensional module which satisfies the ascending chain condition on subdirectly irreducible submodules is Noetherian.
		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 # cabz-15.
	atlas-conferences.com /cgi-bin/abstract/cabz-15 (160 words)

	Ascending chain condition (Site not responding. Last check: 2007-10-20)
		In mathematics, a poset P is said to satisfy the ascending chain condition (ACC) if every ascendingchain a
		The ascending chain condition on P is equivalent to the maximum condition: every nonempty subset ofP has a maximal element.
		Similarly, the descending chain condition is equivalent to the minimumcondition: every nonempty subset of P has a minimal element.
	www.therfcc.org /ascending-chain-condition-329079.html (139 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		ascending chain condition (acc) of md's on varieties of given dimension.
		In particular, this gives the log termination in dimension 3, the special and canonical termination up to dimension 4.
		To prove log termination in dimension 4, one only needs the acc in dimension 4 for the md values in the interval [-1,0].
	www.math.uga.edu /~valery/seminar/shokurov10-29-03.html (107 words)

	ABSTRACT ALGEBRA ON LINE: Ideal Theory of Commutative Rings
		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.
		One important consequence of the generalized principal ideal theorem is that any Noetherian ring satisfies the descending chain condition for prime ideals.
		There may or may not be a uniform bound on the lengths of chains of prime ideals of a Noetherian ring.
	www.math.niu.edu /~beachy/aaol/commutative.html (2296 words)

	Publisher description for Library of Congress control number 94027644 (Site not responding. Last check: 2007-10-20)
		This textbook is affordable and clearly illustrated, and is intended for advanced undergraduate or beginning graduate students with some previous experience of rings and fields.
		Alongside standard algebraic notions such as generators of modules and the ascending chain condition, the book develops in detail the geometric view of a commutative ring as the ring of functions on a space.
		The starting point is the Nullstellensatz, which provides a close link between the geometry of a variety V and the algebra of its coordinate ring A=k[V]; however, many of the geometric ideas arising from varieties apply also to fairly general rings.
	www.loc.gov /catdir/description/cam026/94027644.html (211 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		Subject: Re: abelian groups Date: Mon, 19 Mar 2001 14:15:02 +0100 Newsgroups: sci.math Summary: Finitely generated abelian groups satisfy ascending chain condition.
		druid wrote: > > Can someone please give a hint, I need to show that > finitely generated abelian groups satisfy the > ascending chain condition.
		Since Z, the integers, has ACC, and any f.g.
	www.math.niu.edu /~rusin/known-math/01_incoming/ACC (142 words)

	[No title]
		To show that Xn admits a vn self-map satisfying condition () of Theorem 9 it suffices to exhibit an element v 2 ssR satisfying M pN (4.12.1) k(n)*vp = vn.
		M be a vn self-map satisfying condition (*) of Theorem 9.
		By the ascending chain condition, there is an integer N with the property that Br = BN if r N. But this implies, for r N + 1 that Er EN+1, so image of dr is zero.
	hopf.math.purdue.edu /Hopkins-SmithJH/nilpII.oneside.txt (9081 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		If the set of \htmladdnormallink{annihilators}{http://planetmath.org/encyclopedia/Annihilator.html} $\{ \rann(x) \mid x \in R\}$ satisifies the \htmladdnormallink{ascending chain condition}{http://planetmath.org/encyclopedia/MaximalCondition.html}, then $R$ is said to \htmladdnormallink{satisfy}{http://planetmath.org/encyclopedia/Model.html} the \emph{ascending chain condition on right annihilators}.
		A ring $R$ is called a \emph{right Goldie ring} if it satisfies the ascending chain condition on right annihilators and $R_R$ is a \htmladdnormallink{module of finite rank}{http://planetmath.org/encyclopedia/ModuleOfFiniteRank.html}.
		If the context makes it clear on which side the ring operates, then such a ring is simply called a \emph{Goldie ring}.
	www.ma.utexas.edu /~jcorneli/e/work%20folder/massive/GoldieRing.tex (78 words)

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