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Topic: Ring theory

Related Topics

Commutative ring

Ring (mathematics)

Glossary of ring theory

Noetherian ring

Principal ideal domain

Field (mathematics)

Group (mathematics)

Group theory

Abelian group

Chinese remainder theorem

Natural number

Field theory

Magma (algebra)

Emmy Noether

Laurent series

	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.ebroadcast.com.au /lookup/encyclopedia/ri/Ring_theory.html (531 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 (1890 words)

	[No title]
		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.
		Commutative rings resemble familiar number systems, and various definitions for commutative rings are designed to recover properties known from the integers.
	www.settheory.com /kcell2/wiki_test.html (713 words)

	[Ganoksin] Jewelry Making - Determination of Ring Sizes - Theory and Practice of Goldsmithing -
		In Germany, ring sizes are described by the diameter of the finger hole or inside circumference; in France and the United States, sizes measured by a number, with higher numbers indicating larger sizes.
		Similarly, when a ring size is handed over as a length of string or a mark on a strip of paper, this information must be taken only loosely since those devices have more flexibility than the metal of the final ring.
		Rings with stones cannot be stretched on a mandrel because of the stress this puts on the stone.
	www.ganoksin.com /borisat/nenam/ring_sizes.htm (2489 words)

	Ring Theory: Rings, Ideals, Integral Domains, Fields - Numericana
		The Lord of the Rings by J.R.R. Tolkien (1892-1973).
		Ring of polynomials whose coefficients are in a given ring.
		The radical Rad(I) of an ideal I is the set of all ring elements which have at least one of their powers in I. The radical of an ideal is an ideal.
	home.att.net /~numericana/answer/rings.htm (1316 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/Radical2.html (250 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.
		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.
		Rings associated to group a group G shed light on the structure of G, particular rings of invariants k(V)^G (given a group action on a vector space V), cohomology rings H^(G,Z), group rings* Z[G], and representation rings R(G).
	www.math.niu.edu /~rusin/known-math/index/13-XX.html (2760 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-10-20)
		For example, it is hard to say whether field theory, the theory of finite groups and the theory of finite-dimensional Lie algebras should be regarded as general algebra.
		The appearance of the theory of models and algebraic systems is associated with the discovery of links between algebra and mathematical logic.
		Thus, the concept of a variety of algebras, which crystallized in the theory of universal algebras, plays an important role in modern group and ring theory.
	eom.springer.de /g/g043660.htm (608 words)

	Structure of sub-atomic particles (magnetic ring theory)
		of the ring remains constant and that the ratio of the
		diameter of the ring to the diameter of its cross-section remains constant.
		The magnetic ring theory tells us the reason why it is so.
	www.geocities.com /magneticringtheory (2300 words)

	How the Strong Force fits in Ring Theory
		Ring Theory, as I call the work that I have produced so far, is far from a finished theory.
		The observable effects are L = h angular momentum of a ring, Lspin = h/2pi (measurable 1/2 h/2pi) and E = hw, the energy of the ring itself due to the rotation of the planckons.
		Lspin and Espin are the internal components of a ring and are due to the spinning of the planckons.
	www.mlawrence.co.uk /a42ch301.htm (6303 words)

	Ring Theory as Applied to the Three Classical Geometric Construction Problems
		To show that the three classical construction problems are indeed impossible to solve we must use the theories of Groups, Rings and Fields.
		While the use of the symbols + and * for the two ring operations are intended to parallel our common definitions of addition and multiplication (and guide our intutions to that effect) we should be careful not to make any assumptions about their properties.
		A Field is a commutative ring in which all non-zero elements are units.
	www.pha.jhu.edu /~napora/seminar/r08RingTheory.html (439 words)

	16: Associative rings and algebras
		This includes the study of matrix rings, division rings such as the quaternions, and rings of importance in group theory.
		Solving polynomial equations in the ring of quaternions; passing to extension rings.
		Pointer and citation to solving quadratic equations in the ring of quaternions.
	www.math.niu.edu /~rusin/known-math/index/16-XX.html (541 words)

	Rings (Site not responding. Last check: 2007-10-20)
		A Division Algebra is a nontrivial ring (not necessarily commutative) in which all nonzero elements are invertible.
		The kernel of a homomorphism of rings f: A --> B is the ideal in A consisting of those elements a in A such that f(a) = 0.
		An ideal J in a ring A is prime if the elements of A not in J are closed under multiplication.
	mcraefamily.com /mathhelp/BasicAARings.htm (1006 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)

	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.
		It turns out that the representation theory of groups such as the general linear group and symmetric group is closely connected with Lie theory, through topics like the representation theory of algebraic groups and Lie algebras.
		Typically, the representation theory of such algebras is closely related to the geometry of the prime spectrum of centre of the algebra.
	www.maths.gla.ac.uk /research/groups/algebra/rings.htm (988 words)

	RING THEORY
		It is fascinating to note that the most exciting theory of fundamental particles at the present time, string theory, has a definite resemblance to Thomson's vortex atoms.
		One of the basic entities is the closed string, a little loop, which has fields flowing around it reminiscent of the swirl of ethereal fluid in Thomson's atom.
		There are 2D string theories that incorporate the mechanics of a vortex as two dimensions in a loop which admits to a loop cross-section that has a vortex motion.
	www.astrosciences.info /ringtheory.htm (370 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-10-20)
		He also showed that to construct a cohomology theory it is entirely sufficient to use his canonical flabby resolution, which, from the point of view of homological algebra, turns out to be simply one of the acyclic resolutions of a sheaf.
		Grothendieck and his school vastly generalized sheaf theory, from sheaves on a space to the more general notion of sheaves on a site and that of a topos (cf.
		For the theory of sheaves in the étale topology and for
	eom.springer.de /s/s084840.htm (2238 words)

	Hybrid Ring Duplexer Theory
		The Hybrid Ring Diplexer (duplexer) is a strange device compared to the standard Band Pass or Reject type.
		In common duplexer types, the pass is either created by the band pass effect of a dual coupling loop cavity or the pseudo pass effect of a single loop cavity in BpBr, or the skirt leading up to the notch in a reject type cavity.
		The hybrid ring creates the pass effect from the primary characteristics of a large notch cavity, but you are probably wondering how a notch cavity can create a pass effect.
	www.repeater-builder.com /antenna/hybridring.html (466 words)

	Commutative Ring Theory Seminar, Fall 2002 (Site not responding. Last check: 2007-10-20)
		The goal of the Commutative Ring Theory Seminar is to encourage interaction among members of the department whose interests include commutative ring theory.
		If R is a local ring of characteristic p such that some power of the Frobenius homomorphism f^n: R \to R has finite G-dimension, then R is Gorenstein.
		Serre proved that on an equicharacteristic regular local ring higher Eular characteristics chi(i)(M,N) are non-negative and conjectured that this should hold on any regular local ring.
	www.math.uiuc.edu /~ssather/MATH/crt_fa02.html (737 words)

	GLOSSARY -- Ring Theory as Applied to the Three Classical Geometric Construction Problems
		A ring which is commutative under multiplication is called a Commutative Ring.
		A ring for which the multiplicative identity exists is called a ring with identity.
		If x is an element of a ring R with identity and x has an inverse with respect to * then we call x a unit of R. We denote the set of all units in R by U
	www.pha.jhu.edu /~napora/seminar/Glossary.html (951 words)

	The ring tactic
		In the ring of integers, the normal form of x (3 + yx + 25(1 - z)) + zx is 28x + (-24)xz + xxy.
		Then the tactic guesses the type of given terms, the ring theory to use, the variables map, and replace each term with its normal form.
		The specification of a ring is divided in two parts: first the record of constants (+, ×, 1, 0, -) and then the theorems (associativity, commutativity, etc.).
	flint.cs.yale.edu /cs428/coq/doc/Reference-Manual022.html (2155 words)

	Lee Lady: A Graduate Course in Algebra
		Let a ring R be isomorphic as an R-module to a direct sum of n copies of a left R-module L. Let D be the ring of R-endomorphisms of L. Then L is free as a left D-module with rank n.
		Furthermore, the ring of D-endomorphisms of L is isomorphic to R. (It follows, of course, that R is isomorphic to the opposite ring of the ring of n by n matrices with entries in D.)
		If a ring R has a minimal left ideal L which is not contained in any proper (two-sided) ideal, then R is a finite direct sum of left ideals isomorphic to L and is a simple ring, i.e.
	www.math.hawaii.edu /~lee/algebra/index.html (2759 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)

	The Commutative Ring Theory Webring Home Page (Site not responding. Last check: 2007-10-20)
		The purpose of this webring is to link the home pages of mathematicians and students whose interests include the subject of commutative ring theory, and to do so in such a way that visitors can easily navigate from site to site with minimal time being spent using search engines.
		The answer is simple: a webring is a collection of web pages concerning some common interest (such as commutative ring theory), and these pages are all linked to each other via links such as "Next" and "Previous"; you can see an example of this at the very bottom of this page.
		The organization that keeps track of the behind-the-scenes details, such as the position on the ring of a page being viewed by a browser of the net, is Webring.
	www.math.purdue.edu /~mrogers/webring.html (701 words)

	Bauu Institute Press: Quantum Ring Theory: Foundations for Cold Fusion (Site not responding. Last check: 2007-10-20)
		In Quantum Ring Theory Wladimir Guglinski presents a radical new theory concerning the fundamental nature of physics.
		By considering the nature of “aether” and its role in physical processes, Guglinski is able to put forth a theory that reconciles Quantum Physics with the Theory of Relativity.
		To date, no other physical theory is able to accord for the fundamental intraction between these two fundamental notions of physics.
	www.bauuinstitute.com /Publishing/QuantumBook.html (276 words)

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