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Topic: Commutative ring

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Ring (mathematics)

Glossary of ring theory

Ring theory

Ring ideal

Local ring

Principal ideal

Prime ideal

Module homomorphism

Nilpotent

Integral domain

Principal ideal domain

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Algebra over a field

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	Commutative ring - Wikipedia, the free encyclopedia
		In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law.
		Given a commutative ring R and an ideal I of R, the factor ring R/I is the set of cosets of I together with the operations (a+I)+(b+I)=(a+b)+I and (a+I)(b+I)=ab+I.
		The outer structure of a commutative ring is determined by considering linear algebra over that ring, i.e., by investigating the theory of its modules.
	en.wikipedia.org /wiki/Commutative_ring (831 words)

	Ring (mathematics) - Wikipedia, the free encyclopedia
		Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called commutative rings.
		A ring (in the categorical sense) is commutative iff it is equal to its opposite ring.
		The split-complex plane D is a ring useful in modern physics and is a subring of the tessarines.
	en.wikipedia.org /wiki/Ring_(mathematics) (1102 words)

	Commutative ring -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-09-20)
		If R is a given commutative ring, then the set of all (A mathematical expression that is the sum of a number of terms) polynomials in the variable X whose coefficient are from R forms a new commutative ring, denoted R[X].
		Given a commutative ring R and an (The idea of something that is perfect; something that one hopes to attain) ideal I of R, the (Click link for more info and facts about factor ring) factor ring R/I is the set of cosets of I together with the operations (a+I)+(b+I)=(a+b)+I and (a+I)(b+I)=ab+I.
		The outer structure of a commutative ring is determined by considering linear algebra over that ring, i.e., by investigating the theory of its (A self-contained component (unit or item) that is used in combination with other components) modules.
	www.absoluteastronomy.com /encyclopedia/c/co/commutative_ring.htm (961 words)

	Rings
		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.
		A module over a ring is an abelian group that the ring acts on as a ring of endomorphisms, very much akin to the way fields (integral domains in which every non-zero element is invertible) act on vector spaces.
	www.risberg.ws /Hypertextbooks/Mathematics/Algebra/rings.htm (890 words)

	Commutative ring (Site not responding. Last check: 2007-09-20)
		More generally, every field is a commutative ring, sothe class of fields is a subclass of the class of commutative rings.
		Given a commutative ring R and an ideal I of R,the factor ring R/I is the set of cosetsof I together with the operations (a+I)+(b+I)=(a+b)+I and(a+I)(b+I)=ab+I.
		The outer structure of a commutative ring is determined by considering linear algebra over that ring, i.e., by investigatingthe theory of its modules.
	www.therfcc.org /commutative-ring-76448.html (798 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.
		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.
		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.
	www-groups.dcs.st-and.ac.uk /~history/PrintHT/Ring_theory.html (1857 words)

	PlanetMath: Prüfer ring
		with non-zero unity is a Prüfer ring (cf.
		An additional characterization of Prüfer ring is found here in the entry ``least common multiple'', several other characterizations in [1] (p.
		This is version 81 of Prüfer ring, born on 2004-01-23, modified 2005-08-08.
	planetmath.org /encyclopedia/PruferRing.html (295 words)

	Commutative ring
		The rational, real and complex numbers form commutative rings, in fact they are even fieldss.
		The set of ideals within a commutative ring R can be exactly characterized as the set of R-modules which are subsets of R.
		An non-zero element a of a commutative ring is said to be a zero divisor if there exists another non-zero element b of the ring (b not necessarily distinct from a) so that a*b = 0.
	www.sciencedaily.com /encyclopedia/commutative_ring (886 words)

	[No title] (Site not responding. Last check: 2007-09-20)
		A ring is called commutative if * is also commutative, i.e., if ab = ba for all elements a,b of R. The set of integers {...,-2,-1,0,1,2,...} with the usual operations of addition (+) and multiplication () forms a commutative* ring, and it is the most basic example.
		Commutative rings arose in number theory (as in the Gaussian integers) and were also used in the study of polynomials.
		These days commutative rings are tied very closely to the theory of algebraic geometry, the study of curves, surfaces, etc., defined by polynomial equations.
	www.uwm.edu /~adbell/Research/rtbasics.txt (515 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.
		The greatest early contributor to the theory of non-commutative rings was the Scottish mathematician Wedderburn.
		In 1908 Wedderburn had the important idea of splitting the study of a ring into two parts, one part he called the radical, the part which was left being called semi-simple.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/Ring_theory.html (1857 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)

	Abstracts (Site not responding. Last check: 2007-09-20)
		In the present paper the differential simplicity of a commutative ring is studied with respect to its dimension.
		It is shown that a d-simple ring of prime characteristic is 0-dimensional.
		In the case of characteristic zero a necessary and sufficient condition is given for the d-simplicity of a 1-dimensional finitely generated algebra over a field k and examples are presented of rings with dimension greater than 1 and even of infinite dimension (polynomial rings).
	www.math.unipr.it /~rivista/dati/2001+*/Voskog.html (87 words)

	Commutative ring - Definition up Erdmond.Com
		Given a commutative ring ''R'' and an ideal ''I'' of ''R'', the factor_ring ''R''/''I'' is the set of cosets of ''I'' together with the operations (''a+I'')+(''b+I'')=(''a''+''b'')+I and (''a+I'')(''b+I'')=''ab+I''.
		If ''I'' is an ideal in a commutative ring ''R'', the powers of ''I'' form topological neighborhoods of ''0'' which allow ''R'' to be viewed as a topological_ring.
		A commutative ring with identity which possesses no zero_divisors is called an integral_domain since it closely resembles the integers in some ways.
	www.erdmond.com /Commutative_ring.html (815 words)

	PlanetMath: fractional ideal of commutative ring
		"fractional ideal of commutative ring" is owned by pahio.
		Cross-references: semigroup, principal ideal, ring, factor group, regular, normal subgroup, group, abelian group, contains, finitely generated, invertible, integral, unity, identity element, commutative semigroup, subring, ideal, finite, total ring of fractions, regular element, commutative ring
		This is version 5 of fractional ideal of commutative ring, born on 2005-04-30, modified 2005-07-19.
	planetmath.org /encyclopedia/InvertibleIdeal.html (272 words)

	[No title]
		The category of commutative HZ-algebras (or its weak-unital minor variation) is philosophically the closest category to the cat* egory of E1 ring* spectra with the property that the Hurewicz homomorphism is an iso- morphism.
		For a commutative ring k, the homotopy category of commuta- tive Hk-algebras is equivalent to the homotopy category of E1 differential gra* *ded k-algebras.
		There is a natural isomorphism of commutative k-algebras between the homotopy ring of the commutative Hk-algebra and the homology ring of the corre- sponding E1 differential graded k-algebra.
	hopf.math.purdue.edu /Mandell/hkalg.txt (12848 words)

	Ring (mathematics) (Site not responding. Last check: 2007-09-20)
		The branch of mathematics where rings are studied is called ring theory.
		If a subset S of a ring (R,+,) together with the operations + and restricted on S is itself a ring, and the identity element 1 of R is contained in S, then S is called a subring of (R,+,*).
		Given a ring R and an ideal I of R, the factor ring R/I is the set of cosets of I together with the operations (a+I)+(b+I)=(a+b)+I and (a+I)(b+I)=ab+I.
	www.ukpedia.com /r/ring-mathematics-.html (755 words)

	Ring_Relations
		These assert conditions under which a ring is commutative.
		Commutativity theorems are a special case (we wish to see if xy – yx is a consequence of the identities given in the hypothesis).
		I was able to prove a number of commutativity theorems by choosing, in each case, a collection of substitutions.
	math.ucsd.edu /~jwavrik/web00/Ring_Relations.htm (1136 words)

	PlanetMath: the set of prime ideals of a commutative ring with identity
		The set of all prime ideals of a commutative ring
		"the set of prime ideals of a commutative ring with identity" is owned by bwebste.
		This is version 4 of the set of prime ideals of a commutative ring with identity, born on 2003-10-15, modified 2004-01-07.
	planetmath.org /encyclopedia/SetOfPrimeIdealsOfACommutativeRingWithIdentity.html (111 words)

	ABSTRACT ALGEBRA ON LINE: Rings
		form a class of commutative rings that is a good source of examples and counterexamples.
		S is a commutative ring under componentwise addition and multiplication.
		The ring of all polynomials with real coefficients is also an integral domain, but the larger ring of all real valued functions is not an integral domain.
	www.math.niu.edu /~beachy/aaol/rings.html (1359 words)

	APPENDIX J
		Some elements do; there is a commutative multiplicative group formed by the elements that do have inverses, and the order that group is the value of the Euler function phi(n), which equals the number of integers less than n and greater than zero that are relatively prime to n.
		A division ring R is a ring wherein the elements in R^* the nonzero elements of R, all have multiplicative inverses.
		A Ring Homomorphism is a mapping from a ring R to a ring R' which preserves the ring operations.
	graham.main.nc.us /~bhammel/FCCR/apdxJ.html (5929 words)

	Ring (mathematics) : Commutative ring (Site not responding. Last check: 2007-09-20)
		The branch of mathematics which study rings is called ring theory.
		If a subset H of a ring (R,+,) together with the operations + and restricted on H is itself a ring, then it is called a subring of (R,+,*).
		Given a ring R and and ideal I, the factor ring is the set of cosets of R/I together with operations gI+hI=(g+h)I and (gI)(hI)=ghI.
	www.termsdefined.net /co/commutative-ring.html (782 words)

	[No title]
		Every ring can be viewed as a graded ring with the trivial gradation that assigns degree zero to every element of the ring.
		Fix $g \in G$, and consider the polynomial ring $R[X]$ as a graded extension ring of $R$ uniquely determined by defining $X$ to be a homogeneous element of degree $g$.
		We consider the monoid ring $A[M]$ as a graded ring with its natural $M$-grading where the nonzero elements of $A$ are of degree zero.
	www.math.purdue.edu /~heinzer/preprints/homog42.tex (3790 words)

	Commutative Ring Theory Seminar (Site not responding. Last check: 2007-09-20)
		The goal of this seminar is to encourage interaction among members of the department whose interests include commutative ring theory.
		ABSTRACT: (.dvi or.ps version) In studying Nakayama's 1958 conjecture on rings of infinite dominant dimension, Auslander and Reiten proposed the following generalization: Let $\Lambda$ be an artin algebra and $M$ a $\Lambda$-generator such that $\Ext^i_\Lambda(M,M)=0$ for all $i \geq 1$; then $M$ is projective.
		A modified version is due to Auslander, Ding, and Solberg: Let $\Lambda$ be a Noetherian (possibly noncommutative) ring and let $M$ be a $\Lambda$-module such that $\Ext^i_\Lambda(M \oplus \Lambda, M \oplus \Lambda)=0$ for $i\geq 1$; then $M$ is projective.
	www.math.uiuc.edu /~ssather/MATH/seminar_fa00.html (394 words)

	Rings.
		All the rings above are rings with 1.
		A ring with a 1 will have the characteristic n is and only if n is the smallest
		This is thus a commutative ring with a 1.
	hemsidor.torget.se /users/m/mauritz/math/alg/ring.htm (522 words)

	Commutative Ring Theory (Cambridge Studies in Advanced Mathematics) (Site not responding. Last check: 2007-09-20)
		Commutative Ring Theory (Cambridge Studies in Advanced Mathematics) Review: Matsumura has achieved a great success with this book.
		The first nine chapters contain all of the algebra used in Hartshorne's _Algebraic Geometry_, and a reader who masters all of that material will be well on the way to a solid understanding of algebraic geometry.
		Unfortunately, this book is not sufficient for a first introduction to commutative algebra, as the pace can be brisk for a reader without the mathematical maturity to appreciate Matsumura's condensed style.
	www.textkit.com /0_0521367646.html (381 words)

	The Commutative Ring Theory Webring Home Page (Site not responding. Last check: 2007-09-20)
		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)

	Bublos.com: Compare Book Prices ›› Commutative Ring Theory - Paul-Jean Cahen - Paperback (Site not responding. Last check: 2007-09-20)
		Presenting the proceedings of a recently held conference in Fes, Morocco, this outstanding, up-to-date reference details the latest developments in commutative algebra - highlighting the theory of rings and ideals.
		Exploring commutative algebra's connections with and applications to topological algebra and algebraic geometry, Commutative Ring Theory covers the spectra of rings...chain conditions, dimension theory, and Jaffard rings...fiber products...group rings, semigroups rings, and graded rings...class groups...linear groups...integer-valued polynomials...rings of finite fractions...big Cohen-Macaulay modules...and much more.
		Furnishing over 330 literature citations, allowing further in-depth study of particular topics, Commutative Ring Theory is a vital resource for research mathematicians, algebraists, commutative ring theorists, and graduate students in these disciplines.
	www.bublos.com /isbn/0824791703.html (720 words)

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