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# Topic: Order (ring theory)

 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 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   (1858 words)

 Springer Online Reference Works   (Site not responding. Last check: 2007-10-09) The classical part of the theory of associative rings and algebras is formed by the theory of finite-dimensional associative algebras [2]. The key concepts in the structural theory of associative rings are the concepts of the Jacobson radical, semi-simplicity and primitivity. The concept of the (classical) ring of fractions is important in the structural theory of associative rings. eom.springer.de /a/a013510.htm   (1120 words)

 Ring Theory Although the concept of a ring is due to Dedekind, one of the first words used was an "order" or "order-modul". 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 Wedderburn theory was extended to non-commutative rings satisfying both ascending and descending finiteness conditions (called chain conditions) by Artin in 1927. www-history.mcs.st-andrews.ac.uk /HistTopics/Ring_theory.html   (1890 words)

 Order (ring theory) - Wikipedia, the free encyclopedia In mathematics, an order in the sense of ring theory is a subring O of a ring R that satisfies the conditions In algebraic number theory there are examples for any K other than the rational field of proper subrings of the ring of integers that are also orders. For example, the Hurwitz quaternions are a maximal order in the quaternions with rational co-ordinates; they are not the quaternions with integer coordinates in the most obvious sense. en.wikipedia.org /wiki/Order_(ring_theory)   (357 words)

 PlanetMath: cyclic ring A result of the fundamental theorem of finite abelian groups is that every ring with squarefree order is a cyclic ring. Thus, any infinite cyclic ring that has zero divisors is a zero ring. This is version 27 of cyclic ring, born on 2003-03-10, modified 2006-10-12. planetmath.org /encyclopedia/CyclicRing3.html   (357 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)

 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)

 First-order Model Theory (Stanford Encyclopedia of Philosophy) First-order model theory, also known as classical model theory, is a branch of mathematics that deals with the relationships between descriptions in first-order languages and the structures that satisfy these descriptions. From another point of view, first-order model theory is the paradigm for the rest of model theory; it is the area in which many of the broader ideas of model theory were first worked out. These theories have the remarkable property that every infinite indiscernible sequence in any of their models is indiscernible under any linear ordering whatever; so these sequences are a kind of generalisation of bases of vector spaces. plato.stanford.edu /entries/modeltheory-fo   (6168 words)

 Ideal (ring theory) - Wikipedia, the free encyclopedia (via CobWeb/3.1 planet1.scs.cs.nyu.edu)   (Site not responding. Last check: 2007-10-09) In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The concept of an order ideal in order theory is derived from the notion of ideal in ring theory. + 1 is an ideal in the ring of all polynomials. en.wikipedia.org.cob-web.org:8888 /wiki/Ideal_(ring_theory)   (1237 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)

 Ideal Theory - Cambridge University Press 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. After a discussion of elementary ring theory, he deals with the properties of Noetherian rings and the algebraic and analytical theories of local rings. In order to give some idea of deeper applications of this theory the author has woven into the connected algebraic theory those results which play outstanding roles in the geometric applications. www.cambridge.org /catalogue/catalogue.asp?ISBN=0521604834   (146 words)

 order - OneLook Dictionary Search order, order, order, order, order, order, order, order (of a graph), order (of a group), order (of a group element), order (of zero) : PlanetMath Encyclopedia [home, info] Phrases that include order: limit order, open order, standing order, order of magnitude, pecking order, more... Words similar to order: fiat, edict, arrange, array, club, consecrate, decree, dictate, enjoin, gild, govern, grade, guild, lodge, ordain, orderable, ordered, orderer, ordering, orderless, more... www.onelook.com /cgi-bin/cgiwrap/bware/dofind.cgi?word=order   (833 words)

 Ring_Relations These assert conditions under which a ring is commutative. 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. With the setup as in the first paragraph, we reduce f by g by replacing f by f‑qLgR where q is chosen so that  c-aq is as small as possible in the ordering of integers. math.ucsd.edu /~jwavrik/web00/Ring_Relations.htm   (1136 words)

 [No title] Atiyah and Segal have shown that the complex K-theory ring K*(BG; ZZ) is isomorphic to the I-adic completion of the complex representation ring R(G). In the same section we use a theorem of Kane and Lin to deduce that the p-adic K-theory of a 1-connected p-compact group is an exterior algebra. We write l for the order of ss and embed the latter into the symmetric group l, via the regular representation. www.math.purdue.edu /research/atopology/Jeanneret-Osse/ktheory.txt   (7321 words)

 CATEGORY THEORY AT MCGILL   (Site not responding. Last check: 2007-10-09) What binds them together is that they approach mathematical problems with a point of view that is radically different from that on which traditional mathematics is based, and which emphasizes interactions between mathematical objects over their individual constituents. Their results are often surprising, provide new insights, and are obtained by the invention of sophisticated notions, theories, and techniques. Category Theory at McGill The category theorists that constitute our group are, in order of their joining the Department, Jim Lambek, Marta Bunge, Michael Barr and Michael Makkai,with the addition of Robert Seely and Thomas Fox as Adjunct Professors. www.math.mcgill.ca /bunge/ctatmcgill.html   (549 words)

 Amazon.ca: A First Course in Noncommutative Rings: Books: Tsi-Yuen Lam   (Site not responding. Last check: 2007-10-09) This book, an outgrowth of the author¿s lectures at the University of California at Berkeley, is intended as a textbook for a one-semester course in basic ring theory. A textbook for a one-semester course in basic ring theory. Modern ring theory began when J.H.M. Wedderburn proved his celebrated classification theorem for finite dimensional semisimple algebras over fields. Read the first page www.amazon.ca /First-Course-Noncommutative-Rings/dp/0387953256   (296 words)

 Distance Learning   (Site not responding. Last check: 2007-10-09) ELEC 320 Circuit Theory II First order and second order circuits, natural and forced response, step response, passive and active filters, transformers, dependent sources (modeling, biasing, and gain calculation), Fourier series, Fourier series analysis. Semiclassical laser theory, multimode operation, gas laser theory, ring laser, Zeeman laser. Semi-classical Laser theory, multi-mode operation, gas laser theory, ring laser, Zeeman laser. engineering.alfred.edu /outreach/dist_lrn/index2.html   (752 words)

 The Math Forum - Math Library - Group Theory   (Site not responding. Last check: 2007-10-09) Group theory can be considered the study of symmetry: the collection of symmetries of some object preserving some of its structure forms a group; in some sense all groups arise this way. A book that discusses abelian group theory as not just another variety of group theory but the study of modules over principal ideal domains, a highly specialized branch of commutative ring theory. A paper that presents an integrally generalized theory or conceptualization of life that includes elements of an axiomatic approach and is physically interpretable, formulated as the result of attempts to invent a holistic system of creative synthetic...more>> mathforum.org /library/topics/group_theory   (2240 words)

 The Math Forum - Math Library - Cat. Theory/Homolgcl Alg. A short article designed to provide an introduction to category theory, a comparatively new field of mathematics that provides a universal framework for discussing fields of algebra and geometry. While the general theory and certain types of categories have attracted considerable interest, the area of homological algebra has proved most fruitful in areas of ring theory, group theory, and algebraic topology. The aim of the project is the development of software on a wide variety of platforms for computing with mathematical categories and associated algebraic structures. mathforum.org /library/topics/category_theory   (784 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. mutipliers are related to the dimension theory over the ring of germs of holomorphic functions. it can be generalized over the ring of formal power series with coefficients in any field or over the ring of www.math.uiuc.edu /~jinjiali/MATH/crt2004fall.html   (452 words)

 Amazon.com: Commutative Ring Theory (Cambridge Studies in Advanced Mathematics): Books: H. Matsumura,Miles Reid (via ...   (Site not responding. Last check: 2007-10-09) 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. Commutative Ring Theory (Cambridge Studies in Advanced Mathematics) by H. Plus, get FREE Super Saver Shipping on orders over \$25 shipped from and sold by Amazon.com (restrictions apply). www.amazon.com.cob-web.org:8888 /Commutative-Cambridge-Studies-Advanced-Mathematics/dp/0521367646   (1308 words)

 How to obtain Exploring Abstract Algebra with Mathematica The Springer-Verlag site has a page for our book. Additionally, here are some other links to on-line sources to order this book. Wolfram Research, Inc. (also an entry in their library.wolfram page) www.central.edu /eaam/EAAM/Order.asp   (116 words)

 Theory Ring (Isabelle2005: October 2005) p dvd b)" subsection {* Integral domains *} axclass "domain" < ring one_not_zero: "1 ~= 0" integral: "a * b = 0 ==> a = 0 *) subsection {* Fields *} axclass field < ring field_one_not_zero: "1 ~= 0" (* Avoid a common superclass as the first thing we will prove about fields is that they are domains. *) field_ax: "a ~= 0 ==> a dvd 1" section {* Basic facts *} subsection {* Normaliser for rings *} use "order.ML" method_setup ring = {* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (full_simp_tac ring_ss)) *} {* computes distributive normal form in rings *} subsection {* Rings and the summation operator *} (* Basic facts --- move to HOL!!! www.cl.cam.ac.uk /Research/HVG/Isabelle/dist/library/HOL/HOL-Algebra/Ring.html   (923 words)

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