Ideal-(ring-theory)

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	Ideal (ring theory) - Wikipedia, the free encyclopedia
		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.
		An ideal can be used to construct a factor ring in a similar way as a normal subgroup in group theory can be used to construct a factor group.
		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.
	en.wikipedia.org /wiki/Ring_ideal (1382 words)

	Ring Theory
		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.
		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.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/Ring_theory.html (1857 words)

	Ring theory - InfoSearchPoint.com
		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.
		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.
	www.infosearchpoint.com /display/Ring_theory (659 words)

	PlanetMath: principal ideal ring
		This is version 2 of principal ideal ring, born on 2004-08-23, modified 2004-08-23.
		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
	planetmath.org /encyclopedia/PrincipalIdealRing.html (86 words)

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

	13: Commutative rings and algebras
		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.
		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.
		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)

	PlanetMath: ideal
		(Commutative rings and algebras:: General commutative ring theory :: Ideals; multiplicative ideal theory)
		Cross-references: Dedekind domains, solutions, number fields, number theory, equivalent, commutative ring, subset, ring
		The name ``ideal'' comes from the study of number theory.
	planetmath.org /encyclopedia/Ideal.html (162 words)

	Reference.com/Encyclopedia/Noetherian ring
		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.
		If R is a Noetherian and I is a two-sided ideal, then the factor ring R/I is also Noetherian.
	www.reference.com /browse/wiki/Noetherian_ring (485 words)

	MTH-3E30 : Ring Theory
		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.
		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.
	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)

	Krull
		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.
		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.
		He then wrote the remarkable treatise Ideal Theory which remains a beautiful introduction to ring theory but is simply a theory built from the results that Krull had himself proved.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Krull.html (946 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)

	The Extension and Contraction of an Ideal
		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.
		Map an ideal h into another ring s, and the extension is the linear combinations of all the elements of h mapped into s.
		If the image of a commutative ring r lies in the center of s, the contraction of a prime/semiprime ideal is same.
	www.mathreference.com /ring,conext.html (383 words)

	Ring_Relations
		In his Carus monograph, Noncommutative Rings, Herstein shows how general versions of some commutativity theorems can be obtained using the structure theory for rings.
		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.
		These assert conditions under which a ring is commutative.
	math.ucsd.edu /~jwavrik/web00/Ring_Relations.htm (1136 words)

	Prime Ideal Correspondence
		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)

	Open Questions: Algebraic Number Theory
		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.
		Rings of algebraic integers are what algebraic number theory is all about.
		A proper ideal of R is an ideal I that is not equal to R, i.
	www.openquestions.com /oq-ma018.htm (19624 words)

	ADVANCES IN RING 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.
		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.
		Also included are original articles showing the trend of current research.
	www.worldscibooks.com /mathematics/5903.html (175 words)

	research.html
		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 $R=k[x_1,\dots,x_n]$.
		Commutative ring theory is an essential ingredient of algebraic geometry.
		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.
	dreadnought.uta.edu /~dave/research.dir/research.html (1786 words)

	OUOSU ring theory seminar speakers for 2004-5
		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.
		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.
		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 : Eric Weisstein's World of Mathematics [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)

	Commutative Ring Theory Seminar, Spring 2004
		The goal of the Commutative Ring Theory Seminar is to encourage interaction among members of the department whose interests include commutative ring theory.
		Speakers will discuss current work in commutative ring theory.
		Let R be a local ring of depth d and M a finite R-module.
	www.math.uiuc.edu /~ssather/MATH/crt.html (661 words)

	Amazon.com: Commutative Ring Theory (Cambridge Studies in Advanced Mathematics): Books
		In addition to being an interesting and profound subject in its own right, commutative ring theory is important as a foundation for algebraic geometry and complex analytical geometry.
		The Algebraic Theory of Modular Systems (Cambridge Mathematical Library) by F. Macaulay in Front Matter (1), and Front Matter (2)
		More advanced topics such as Ratliff's theorems on chains of prime ideals are also explored.
	www.amazon.com /exec/obidos/tg/detail/-/0521367646?v=glance (936 words)

	11R: Algebraic number theory: global fields
		The rings of integers in these fields are also studied in Commutative Ring Theory, where we may find discussion of Unique Factorization Domains (UFDs), Euclidean domains, etc.
		Pohst, M.; Zassenhaus, H.: "Algorithmic algebraic number theory", Encyclopedia of Mathematics and its Applications, 30.
		There are quite a few texts in algebraic number theory.
	www.math.niu.edu /~rusin/known-math/index/11RXX.html (612 words)

	Commutative Ring Theory RAP
		Assume that (R,m) is a regular local ring containing a field and that p and q are prime ideals of R such that (i) the ideal p+q is primary to the maximal ideal m, and (ii) dim(R/p)+dim(R/q)=dim(R).
		During the Spring of 2001, the commutative ring theory group will continue its RAP (Research Among Peers).
		Tuesday, 16 January: Organizational meeting for Commutative Ring Theory Seminar and RAP.
	www.math.uiuc.edu /~ssather/MATH/RAP_sp01.html (610 words)

	- SHOP.COM
		Ideal theory is important not only for the intrinsic interest and purity of its logical structure but because it is a necessary tool in many branches of mathematics.
		In this introduction to the modern theory of ideals, Professor Northcott assumes a...
		You might try modifying your search term or selecting one of the department links below.
	www.shop.com /op/aprod-p26460652 (240 words)

	Business Software Review : Article 'Combined systems'
		glossary of field theory glossary of group theory glossary of ring theory glossary of tensor theory.
		See also: list of commutative algebra topics list of group theory topics list of homological algebra topics list of linear algebra topics.
		Symbolic mathematics Finite field arithmetic Grýbner basis Buchberger's algorithm Axiom computer algebra system, Axiom (algebra software) GAP computer algebra system Ginac computer algebra system Macaulay computer algebra system Magma computer algebra system Maple computer algebra system Mathematica Maxima computer algebra system PARI-GP computer algebra system Reduce computer algebra system SINGULAR computer algebra system Yacas
	www.business-software-review.org /DisplayArticle1222564.html (8712 words)

	Ring Theory - Dictionary of Mathematics
		Dictionary > Science > Mathematics > Ring Theory
	www.dictionaryofeverything.com /mathematics/ring_theory.html (8 words)

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Topic: Ideal-(ring-theory)

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In the News (Wed 30 Dec 09)