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Topic: Ideal (ring theory)

	Ideal (ring theory) - Wikipedia, the free encyclopedia
		In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring.
		The map p from R to R/I defined by p(a) = a + I is a surjective ring homomorphism (or ring epimorphism) whose kernel is the original ideal I.
		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.
	en.wikipedia.org /wiki/Ring_ideal (1382 words)

	Ring theory: Definition and Links by Encyclopedian.com - All about Ring theory
		In mathematics, Ring theory is that branch of mathematics concerned with the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers.
		The theory of commutative rings resembles the theory of numbers in several respects, and various definitions for commutative rings are designed to recover properties known from the integers.
		In commutative ring theory, numbers are often replaced by ideals, and the definition of prime ideal tries to capture the essence of prime numbers.
	www.encyclopedian.com /ri/Ring-theory.html (562 words)

	Ideal (ring theory) - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-10-13)
		In ring theory, a branch of abstract algebra, an ideal of a ring R is a subset I of R which is closed under R-linear combinations, in a sense made precise below.
		The term "ideal" comes from the notion of ideal number: ideals were seen as a generalization of the concept of number.
		In the ring Z of integers, every ideal can be generated by a single number (so Z is a principal ideal domain), and the ideal determines the number up to its sign.
	encyclopedia.learnthis.info /i/id/ideal__ring_theory_.html (1381 words)

	Ring Theory
		However, axioms for rings are not given by Weber and the axiomatic treatment of commutative rings was not developed until the 1920's in the work of Emmy Noether and Krull.
		In contrast to commutative ring theory, which as we have seen grew from number theory, non-commutative ring theory developed from an idea which, at the time of its discovery, was heralded as a great advance in applied mathematics.
		The greatest early contributor to the theory of non-commutative rings was the Scottish mathematician Wedderburn.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/Ring_theory.html (1857 words)

	Ring Theory
		In this article we shall be concerned with the development of the theory of commutative rings (that is rings in which multiplication is commutative) and the theory of non-commutative rings up to the 1940's.
		Ring theory in its own right was born together with an early hint of the axiomatic method which was to dominate algebra in the 20
		It is important to realise that at this stage rings of polynomials and rings of numbers were being studied, but it was to be another 40 years before an axiomatic theory of commutative rings was to be developed bringing these theories together.
	www-history.mcs.st-and.ac.uk /PrintHT/Ring_theory.html (1857 words)

	PlanetMath: ideal
		When the failure of unique factorization in number fields was first noticed, one of the solutions was to work with so-called ``ideal numbers'' in which unique factorization did hold.
		These ``ideal numbers'' were in fact ideals, and in Dedekind domains, unique factorization of ideals does indeed hold.
		This is version 11 of ideal, born on 2001-10-19, modified 2005-01-17.
	planetmath.org /encyclopedia/Ideal.html (162 words)

	Ring theory - InfoSearchPoint.com (Site not responding. Last check: 2007-10-13)
		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.
		The study of rings originated from the theory of polynomial rings and the theory of algebraic integers.
		In 1921, Emmy Noether gave the first axiomatic foundation of the theory of commutative rings in her monumental paper Ideal Theory in Rings.
	www.infosearchpoint.com /display/Ring_theory (659 words)

	PlanetMath: radical theory
		Rings are not required to have an identity element in radical theory.)
		Radical theory is the study of radical properties and their interrelations.
		This is version 7 of radical theory, born on 2002-12-07, modified 2004-02-28.
	planetmath.org /encyclopedia/RadicalTheory.html (251 words)

	13: Commutative rings and algebras
		Of particular interest are several classes of rings of interest in number theory, field theory, algebraic geometry, and related areas; however, other classes of rings arise, and a rich structure theory arises to analyze commutative rings in general, using the concepts of ideals, localizations, and homological algebra.
		Ideals may be added, multiplied, or intersected, giving a sort of combinatorial structure to the set of ideals in a ring; in particular, the prime ideals play a special role among the set of ideals in this way.
		Conversely, the study of a ring is often focused by the examination of related fields, such as the quotients by each of the maximal ideals, or, in the case of integral domains, by the quotient field.
	www.math.niu.edu /~rusin/known-math/index/13-XX.html (2760 words)

	Reference.com/Encyclopedia/Noetherian ring
		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.reference.com /browse/wiki/Noetherian_ring (485 words)

	PlanetMath: principal ideal ring
		generated by a single ring element, is called a principal ideal ring.
		Cross-references: field, polynomial ring, factor rings, integers, principal ideal domain, integral domain, ring, generated by, ideals, commutative ring
		This is version 2 of principal ideal ring, born on 2004-08-23, modified 2004-08-23.
	planetmath.org /encyclopedia/PrincipalIdealRing.html (86 words)

	Krull (Site not responding. Last check: 2007-10-13)
		Ring theory results from this thesis have recently been found important in the area of coding theory.
		In 1928 he defined the Krull dimension of a commutative Noetherian ring and brought ring theory into in new setting in which he was able to show that the principal ideal theorem held.
		Another major topic in ring theory is the study of local rings, that is rings having a unique maximal ideal, and they are used in the study of local properties of algebraic varieties.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Krull.html (946 words)

	The Extension and Contraction of an Ideal (Site not responding. Last check: 2007-10-13)
		If f is a ring homomorphism from r into s, the extension of an ideal in r is the ideal generated by its image in s, and the contraction of an ideal in s is its preimage in r, which is already an ideal.
		If the image of a commutative ring r lies in the center of s, the contraction of a prime/semiprime ideal is same.
		Map an ideal h into another ring s, and the extension is the linear combinations of all the elements of h mapped into s.
	www.mathreference.com /ring,conext.html (383 words)

	MTH-3E30 : Ring Theory
		In the course the basic results of non-commutative ring theory will be proved and the following topics are included: simple and semi-simple rings, division rings, radicals and the elementary theory of modules.
		Overview: Ring theory has many applications in mathematics and thanks to the re-discovery of its connections to physics in the last fifteen years is playing an ever more important role.
		One of the most powerful and efficient methods in non-commutative ring theory is the analysis of their representations.
	www.mth.uea.ac.uk /maths/syllabuses/9900/3E3001.html (335 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/9900/2A2400.html (521 words)

	Open Questions: Algebraic Number Theory
		Galois theory is a way to "map" extensions of fields to groups and their subgroups in such a way that most of the interesting details about the extension are reflected in details about the groups, and vice versa.
		A proper ideal of R is an ideal I that is not equal to R, i.
		On the other hand, class field theory provides a complete answer of a different kind which specifies a 1-to-1 correspondence between abelian extensions of a number field and certain generalized ideal class groups which are defined entirely from the arithmetic properties of the base field.
	www.openquestions.com /oq-ma018.htm (19624 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.
		is a commutative ring with identity and the polynomial ring
		This is the converse of a well-known theorem.
	www.math.dartmouth.edu /graduate-students/syllabi/sample-questions/algebra/node3.html (274 words)

	Prime Ideal Correspondence (Site not responding. Last check: 2007-10-13)
		This is similar to the ideal correspondence theorem, but now we are dealing with prime ideals.
		Therefore an ideal is prime iff its image or preimage is prime.
		The ideal p, generated by y, is no longer prime, since it contains x×x, and x is not in p.
	www.mathreference.com /ring,primcor.html (203 words)

	Lee Lady: Finite Rank Torsion Free Modules over Dedekind Domains (a book)
		Kaplansky, in his "little red book", asserted that abelian group theory is really the study of modules over principal ideal domains, and since then most abelian group theorists tend to feel more at home with commutative ring theory than with group theory in general.
		The theory of finite rank torsion free abelian groups is full of results that depend on countability, or on having characteristic zero, or working over a ring whose quotient field is a perfect field, as well as proofs using quite specialized results from number theory.
		Unlike the theory of torsion groups, the theory of finite rank torsion free modules is becoming something that fits in fairly well with the mainstream of commutative ring theory.
	www.math.hawaii.edu /~lee/book (629 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		This leads to what has come to be known in commutative ring theory as a free resolution of the ideal $I=(f_1,\dots,f_m)$ over the polynomial ring $R=k[x_1,\dots,x_n]$.
		In theory, all of the information about a given resolution is present in the first step; the challenge lies in knowing how to extract it.
		Huneke and I introduced {\it AB rings\/}, which are rings whose modules satisfy the following strong form of a conjecture of Auslander: if $\Ext^i_R(M,N)=0$ for all $i\gg 0$, then they vanish for all $i$ greater than some fixed integer $n$ depending on the ring $R$.
	dreadnought.uta.edu /~dave/research.dir/research.html (1786 words)

	Ring_Relations
		These assert conditions under which a ring is commutative.
		the ring has a 1 or there are no non-zero nilpotents) The desired conclusion is that the identity xy-yx holds.
		The set of consequences of a set of identities has the same arithmetic closure properties of an ideal – but, in addition, it is closed under the operation of substitution of polynomials for its variables.
	math.ucsd.edu /~jwavrik/web00/Ring_Relations.htm (1136 words)

	Commutative Ring Theory Seminar, Spring 2004 (Site not responding. Last check: 2007-10-13)
		The goal of the Commutative Ring Theory Seminar is to encourage interaction among members of the department whose interests include commutative ring theory.
		Let R be a local ring of depth d and M a finite R-module.
		If the module M/IM has finite G-dimension for every ideal I generated by a maximal R-sequence, then either depth(M) \geq d or R is Gorenstein.
	www.math.uiuc.edu /~ssather/MATH/crt.html (661 words)

	OUOSU ring theory seminar speakers for 2004-5
		November 5, 2004: Warrem McGovern, Bowling Green StateUniversity, Rings of quotients of C(X) ABSTRACT: Q(X) and q(X) denote the maximal and classical rings of quotients of C(X) (resp.) It is known that, since C(X) is a semiprime ring, Q(X) is always von-Neumann regular.
		Inspired by a classic treatment of Ore [1932], a consideration of the arithmetic of such a ring \(R \) on an element-by-element basis (rather than by an ideal-theoretic analysis) leads to a very nice description of the internal structure of the indecomposables.
		The work is not complete: there is a detailed analysis of the structure of the endomorphism rings of the indecomposables underway; and some general questions about the nature of localizations intermediate between the first Weyl algebra and the Weyl division algebra.
	www.math.ohiou.edu /~lopez/log0405.html (826 words)

	IDEAL - OneLook Dictionary Search
		Ideal, ideal : UltraLingua English Dictionary [home, info]
		Phrases that include IDEAL: ideal gas, ego ideal, beau ideal, ideal gas law, ideal solid, more...
		Words similar to IDEAL: apotheosis, edenic, idealistic, idealless, nonesuch, nonpareil, nonsuch, paragon, saint, utopian, best, dream, exemplar, model, paradigm, perfect, standard, unparalleled, more...
	www.onelook.com /cgi-bin/cgiwrap/bware/dofind.cgi?word=IDEAL (396 words)

	ADVANCES IN RING THEORY
		The selected papers in this volume cover all the most important areas of ring theory and module theory such as classical ring theory, representation theory, the theory of quantum groups, the theory of Hopf algebras, the theory of Lie algebras and Abelian group theory.
		The review articles, written by specialists, provide an excellent overview of the various areas of ring and module theory — ideal for researchers looking for a new or related field of study.
		Also included are original articles showing the trend of current research.
	www.worldscibooks.com /mathematics/5903.html (175 words)

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