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Topic: Artinian ring

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	unit (algebra) (Site not responding. Last check: 2007-10-08)
		Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation.
		A ring satisfying the descending chain condition for left ideals is left artinian; if it satisfies the descending chain condition for right ideals, it is right artinian; if it is both left and right artinian, it is called artinian.
		A ring satisfying the ascending chain condition for left ideals is left noetherian; a ring satisfying the ascending chain condition for right ideals is right noetherian; a ring that is both left and right noetherian is noetherian.
	www.yourencyclopedia.net /Unit_(algebra).html (1177 words)

	Artinian - Wikipedia, the free encyclopedia
		In mathematics, Artinian is an adjective that describes objects that satisfy particular cases of the descending chain condition.
		A ring is an Artinian ring if it satisfies the descending chain condition on ideals.
		The concept is named for Emil Artin, who classified all simple rings whose one-sided ideals satisfy the descending chain condition.
	en.wikipedia.org /wiki/Artinian (133 words)

	Chiusura integrale: Tutte le informazioni su Chiusura integrale su Encyclopedia.it (Site not responding. Last check: 2007-10-08)
		Rings of polynomials are integral domains if the coefficients come from an integral domain.
		For instance, the ring Z[X] of all polynomials in one variable with integer coefficients is an integral domain; so is the ring R[X,Y] of all polynomials in two variables with real coefficients.
		The same is true for rings of analytical functions on connected open subsets of analytical manifolds.
	www.encyclopedia.it /c/ch/chiusura_integrale.html (654 words)

	[No title]
		Artinian ring +------------------------------------------------------------ An Artinian ring is a ring which when considered as a R-module is an Artinian module.
		Artinian ring +------------------------------------------------------------ Two elements of an integral domain that are unit-multipliers of each other are called associate numbers.
		ring +------------------------------------------------------------ A ring (X,+,,0) is a set X with a binary operation + and a binary operation such that (X,+,0) is a commutative group and (X,) is a semigroup and such that the distributivity laws a(b+c) = ab + ac, (a+b)c - ac+b*c hold.
	www.math.harvard.edu /~knill/sofia/data/algebra.txt (1599 words)

	Jacobson density theorem - Wikipedia, the free encyclopedia
		In mathematics, the Jacobson density theorem in ring theory is an important generalization of the Artin-Wedderburn theorem.
		That this is a division ring is a consequence of Schur's lemma.
		In particular, when the ring is primitive then it is a dense subring of linear transformations of a vector space over a division ring.
	en.wikipedia.org /wiki/Jacobson_density_theorem (178 words)

	Artinian ring - Wikipedia, the free encyclopedia
		In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals.
		The Artin-Wedderburn theorem characterizes all simple rings that are Artinian: they are the matrix rings over a division ring.
		By the Hopkins-Levitzski theorem, a left (right) Artinian ring is automatically a left (right) Noetherian ring.
	en.wikipedia.org /wiki/Artinian_ring (213 words)

	Glossary of ring theory (Site not responding. Last check: 2007-10-08)
		; Artinian ring : A ring satisfying the descending chain condition for left ideals is left artinian ; if it satisfies the descending chain condition for right ideals, it is right artinian ; if it is both left and right artinian, it is called artinian.
		Field theory is in fact an older branch of mathematics than ring theory.
		; Noetherian ring : A ring satisfying the ascending chain condition for left ideals is left noetherian ; a ring satisfying the ascending chain condition for right ideals is right noetherian ; a ring that is both left and right noetherian is noetherian.
	www.serebella.com /encyclopedia/article-Glossary_of_ring_theory.html (1825 words)

	Noetherian ring (Site not responding. Last check: 2007-10-08)
		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".
		The Noetherian property is central in ring theory and in areas that make heavy use of rings, such as algebraic geometry.
	www.tocatch.info /en/Noetherian_ring.htm (457 words)

	Glossary of ring theory
		; Irreducible : An element x of a ring is irreducible if for any elements a and b such that x=a b, either a or b is a unit.
		; Prime : An element x of a ring is prime if for any elements a and b such that x=a b, either x divides a or x divides b.
		To localize a ring R, take a multiplicatively closed subset S containing no zero devisors, and formally define their inverses, which shall be added into R.
	www.sciencedaily.com /encyclopedia/glossary_of_ring_theory (1579 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		Corollary.} {For a ring $R$ the following are equivalent: \itemitem{} {\rm (a)} $R$ is simple and left artinian; \itemitem{} {\rm (b)} $R \cong \fatm_n(D)$ for some $n\in \fatn$ and some division ring $D$; \itemitem{} {\rm (c)} $R$ is simple and right artinian.
		So by Exercise 9, the ring $R =\End(V_D)$ of operators on $V$ is isomorphic to the ring $\colfin_\fatn(D)$ of all $\fatn\times \fatn$ column finite matrices over $D$.
		Prove that $R$ is semi--primitive iff there is an injective ring homomorphism $\phi:R \longrightarrow \prod_A S_\alpha$ where each $S_\alpha$ is the ring of operators $\End(V_\alpha)$ on a right vector space $V_\alpha$ over a division ring $D_\alpha$, and for each alpha the image $\pi_\alpha\phi(R)$ is dense in $S_\alpha$.
	darkwing.uoregon.edu /~anderson/math648/lecture19.html (1721 words)

	PlanetMath: composition series
		A necessary and sufficient condition for a module to have a composition series is that it is both Noetherian and Artinian.
		If a module does have a composition series, then all composition series are the same length.
		This is version 1 of composition series, born on 2003-11-22.
	planetmath.org /encyclopedia/CompositionSeries3.html (121 words)

	UWM Math: Noetherian Rings (Site not responding. Last check: 2007-10-08)
		The main thrust of the theory of commutative rings is intimately related to the theory of rings of polynomial functions (and rings derived from them such as quotients and localizations).
		The study of non-commutative rings is a field begun in the 20th century, and much of the early work concentrated on division rings and algebras that were finite dimensional over a field.
		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)

	Finite Rings (Site not responding. Last check: 2007-10-08)
		An element of a ring that does not lie in any maximal ideal is a unit (by applying Zorn’s lemma; however, this is not required for finite rings or finite dimensional algebras).
		The Nilpotency of the Nil radical of a Noetherian Ring.
		The Jacobson radical of an Artinian ring is the product of its (finite collection of) maximal ideals and is a nilpotent ideal; each prime ideal is maximal and consists of zero-divisors; the complement of the union of all maximal ideals consists of units.
	www.imsc.ernet.in /~kapil/geometry/caag/finite.html (2519 words)

	ABSTRACT ALGEBRA ON LINE: Structure of Noncommutative Rings
		The ring R is called a simple ring if (0) is a maximal ideal; it is called a prime ring if (0) is a prime ideal, and a semiprime ring if (0) is a semiprime ideal.
		In a left Artinian ring, the notions of maximal ideal, primitive ideal, and prime ideal coincide.
		If P is a primitive ideal of the ring R, then there exists a division ring D and a vector space V over D for which R/P is isomorphic to a subring of the ring of all linear transformations from V into V. Proposition.
	www.math.niu.edu /~beachy/aaol/noncommutative.html (1031 words)

	Noetherian ring - Term Explanation on IndexSuche.Com
		The Noetherian property is central in ring_theory and in areas that make heavy use of rings, such as algebraic_geometry.
		As another application, we mention one in a commutative Noetherian ring is a principal_ideal.
		The ring of polynomials in infinitely-many variables, ''X1'', ''X2'', ''X3'', etc. The sequence of ideals (''X1''), (''X1'',''X2''), (''X1'',''X2'', ''X3''), etc. is ascending, and does not terminate.
	www.indexsuche.com /Noetherian_ring.html (493 words)

	Artinian module - InformationBlast
		In abstract algebra, an Artinian module is a module that satisfies the descending chain condition on its submodules.
		When working with an Artinian ring we must distinguish between being left, right, or two-sided Artinian, but this distinction does not make sense when working with modules.
		Unlike the case of rings, there are Artinian modules which are not Noetherian modules.
	www.informationblast.com /Artinian_module.html (159 words)

	Artinian ring (Site not responding. Last check: 2007-10-08)
		Emil Artin first discovered that the descending chain for ideals generalizes both classes of rings Artinian rings are named after him.
		A ring is Artinian or two-sided Artinian if it is both left and Artinian.
		The Lord of the Rings: The Fellowship of the Ring
	www.freeglossary.com /Artinian_ring (565 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		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)

	Primitive ring - Wikpedia (Site not responding. Last check: 2007-10-08)
		In abstract algebra, a left primitive ring R is a ring with a faithful simple left module R-module.
		An Artinian ring is primitive if and only if it is simple.
		A commutative ring is primitive if and only if it is a field.
	www.bostoncoop.net /~tpryor/wiki/index.php?title=Primitive_ring (60 words)

	Le résultat de votre recherche
		This book is the first to present a complete theory of filtrations on associative rings, combining techniques stemming from number theory related to valuations, with facts originating in the study of rings of differential operators on varieties.\par It is divided into four chapters, each of which is subdivided into sections.
		The case where the associated graded ring of a filtered simple Artinian ring is a semiprime P.I. ring is reduced to the prime case, by using microlocalization; without P.I. hypothesis an extra assumption is necessary in order to arrive at the same conclusion (see resp.
		Zariski rings with noetherian Rees ring and with commutative associated graded ring are investigated.
	www.math.jussieu.fr /~keller/semalg/li.html (2114 words)

	AR-theory for Artinian PI-Rings (Site not responding. Last check: 2007-10-08)
		Let R be an artinian polynomial identity ring, that is an artinian ring whose radical factor is an artin algebra.
		Let R be an artinian PI-ring and M a finitely generated indecomposable R-module.
		Let R be an artinian PI-ring, C a connected component of the Auslander-Reiten quiver and M in C.
	www.math.fau.edu /Schmidme/AR-theory.html (326 words)

	Selected Matches for: Items Authored by Clark, W. Edwin
		For elements of a Euclidean ring $R$ with a real valuation $v(R)$, the authors establish a new canonical form, which generalizes previously known forms, that is valid for integers with an arbitrary radix $r$ and for Gaussian integers with radix $r=±1±i$.
		A ring with 1 is called a chain ring if its lattice of left ideals forms a chain, and it is known that a finite chain ring is a local uniserial ring and the lattice of its right ideals forms a chain, as well.
		Univ., Kazan, 1964; MR 34 #190] showed that a finite chain ring is a homomorphic image of a ring of the form $\langle x,y\rangle/I$, where $\langle x,y\rangle$ is the ring of polynomials in the non-commuting indeterminates $x,y$ over $Z\sb {p\sp n}$ and $I$ is an ideal generated by four elements of specified type.
	www.math.usf.edu /~eclark/pubs_mathscinet.html (6661 words)

	Artinian (Site not responding. Last check: 2007-10-08)
		In mathematics, a ring is Artinian if it has the descending chain condition on the poset of ideals under inclusion.
		An algebra over a field that is finite-dimensional (over the field) is certainly Artinian, since any ideal must be a vector subspace.
		Therefore the theory of Artinian rings (named for Emil Artin) has many classical examples.
	www.theezine.net /a/artinian.html (74 words)

	Citations: Artinian subrings of a commutative ring - Gilmer, Heinzer (ResearchIndex) (Site not responding. Last check: 2007-10-08)
		Citations: Artinian subrings of a commutative ring - Gilmer, Heinzer (ResearchIndex)
		Gilmer and W. Heinzer, Artinian subrings of a commutative ring, Trans.
		....Artinian case, two key properties of an Artinian ring R that come into play are that Spec(R) is nite and that R has only Commutative Rings of Dimension 0 7 nitely many idempotents.
	citeseer.ist.psu.edu /context/222871/0 (475 words)

	OhioLINK ETD: Er, Noyan (Site not responding. Last check: 2007-10-08)
		A Module M is called a CS module if every submodule of M is essential in a direct summand of M. In this dissertation certain classes of rings characterized by direct sums of CS modules are considered.
		(N) is CS, and; (ii) R is a right Artinian ring and all uniform right R-modules have composition length at most two iff the direct sum of any two CS right R-modules is again CS.
		Partial answers are obtained to a question of Huynh whether a semilocal ring or a ring with finite right uniform dimension that is right countably Σ-CS is right Σ-CS.
	www.ohiolink.edu /etd/view.cgi?ohiou1069450867 (218 words)

	Chains of Modules (Site not responding. Last check: 2007-10-08)
		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.
		A division ring is a rather trivial example.
	www.mathreference.com /mod-acc,intro.html (236 words)

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