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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. 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. Examples of non-commutative rings are given by rings of square matrices or more generally by rings of endomorphisms of abelian groups or modules, and by monoid rings. www.risberg.ws /Hypertextbooks/Mathematics/Algebra/rings.htm   (890 words)

ideal   (Site not responding. Last check: 2007-11-06) In 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. 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. For instance, in general rings one studies prime ideals instead of prime numbers, one defines coprime ideals as a generalization of coprime numbers, and one can prove a generalized Chinese remainder theorem about ideals. www.yourencyclopedia.net /Ideal.html   (1346 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, 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   (84 words)

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. Also, the union of two ideals is a subset of the sum of those two ideal. en.wikipedia.org /wiki/Ideal_(ring_theory)   (1343 words)

ABSTRACT ALGEBRA ON LINE: Rings S is a commutative ring under componentwise addition and multiplication. that arise in the prime factorization of n. 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)

Factor rings and the isomorphism theorems   (Site not responding. Last check: 2007-11-06) Note that the zero of the factor ring is the coset 0 + I or the ideal I itself. The way you should think of the factor ring R/I is that it is the ring R in which "all the elements in the ideal I have been made into zero". We saw in the last section that the kernel of a ring homomorphism is an ideal and the image is a subgroup. www-groups.dcs.st-and.ac.uk /~john/MT4517/Lectures/L8.html   (906 words)

Polynomial - Wikipedia, the free encyclopedia One can then check that the set of all polynomials with coefficients in the ring R forms itself a ring, the ring of polynomials over R, which is denoted by R[X]. Formation of the polynomial ring, together with forming factor rings by factoring out ideals, are important tools for constructing new rings out of known ones. The reason that algebraists have to distinguish between polynomials and polynomial functions is that over some rings R (for instance, over finite fields), two different polynomials may give rise to the same polynomial function. en.wikipedia.org /wiki/Polynomial   (2063 words)

UWM Math: Noetherian Rings   (Site not responding. Last check: 2007-11-06) The main thrust of the theory of commutative rings is intimately related to the theory of rings of polynomial functions (and rings derived from them such as quotients and localizations). The study of non-commutative rings is a field begun in the 20th century, and much of the early work concentrated on division rings and algebras that were finite dimensional over a field. While many interesting ring theoretic results were proven in between, it is probably fair to say that the modern study of non-commutative noetherian rings began with A. Goldie's work in 1958-1960 giving necessary and sufficient conditions for a ring to have a semisimple ring of fractions. www.uwm.edu /Dept/Math/Research/Algebra/noetherian/noetherian.html   (467 words)

class rings   (Site not responding. Last check: 2007-11-06) A subset I of the ring R is a left ideal of R... Ring homomorphisms Let R and S be ringss and let f be a ring homomorphism from R to S. If 0S is... One usually thinks of R as the ring of integers in F. It is a discrete valuation ring with quotient field F. If F is Qp, then R is the ring of p-adic integers Zp; if F is... www.byglrb.com /jewelry/class+rings   (1438 words)

Polynomial -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-11-06) are elements of some (Jewelry consisting of a circlet of precious metal (often set with jewels) worn on the finger) ring R and X is considered to be a formal symbol. Formation of the polynomial ring, together with forming factor rings by factoring out (The idea of something that is perfect; something that one hopes to attain) ideals, are important tools for constructing new rings out of known ones. Analogously, polynomial "primes" (more correctly, (Click link for more info and facts about irreducible) irreducible polynomials) can be defined which cannot be factorized into the product of two polynomials of lesser (A specific identifiable position in a continuum or series or especially in a process) degree. www.absoluteastronomy.com /encyclopedia/p/po/polynomial.htm   (2252 words)

[No title]   (Site not responding. Last check: 2007-11-06) Part of the attraction of 2-primal rings (in addition to their being a common generalization of commutative rings and rings without nilpotent elements) lies in the structure of their prime ideals. In the investigation of skew polynomial rings, the 2-primal condition was studied via Shin's minimal-prime criterion, based on an understanding of extension and contraction of primes between the base ring the the skew polynomial extension. Because the factor ring $R/\mathfrak{m}$ is a domain and the endomorphism $\sigma'$ is injective, Equation~(\ref{Oreisomorphism}) implies that the factor ring $S/\mathfrak{m}S$ is a domain, i.e.\ that the ideal $\mathfrak{m}S \subset S$ is completely prime. euler.slu.edu /Dept/Faculty/marks/On2PrimalOreExtensions.txt   (2911 words)

[No title] The aim of the course is to introduce students to the basic abstract algebraic structures such as groups, rings and fields. Rings: Integral domains, homomorphisms, ideals, factor rings, polynomial rings, factorization of polynomials. Fields: Finite extensions and their connection with polynomial rings, construction of finite extensions, application to geometric constructibility problems. www.math.uic.edu /~brayton/330   (570 words)

[No title] abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. fundamental theorem of arithmetic: in these rings, every nonzero ideal can be uniquely written as a product of prime ideals. 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 en-cyclopedia.com /wiki/Maximal_ideal   (1185 words)

Abstract algebra:Ideals - Wikibooks The development can be done with non-commutative rings but it is more complicated than is necessary at the moment. To verify that a subset of a commutative ring is an ideal, it is only necessary to check that it closed under subtraction and that it absorbs multiplication. Definition/Theorem: This set of cosets, called the factor ring (or quotient ring) of R modulo I is a ring with operations en.wikibooks.org /wiki/Abstract_algebra:Ideals   (582 words)

Coagulation Factor Xa Induces Endothelium-Dependent Relaxations in Rat Aorta -- Schaeffer et al. 81 (5): 824 -- ... Effect of thrombin (TH), factor Xa (Xa), and trypsin (Tryp) on phenylephrine (PE, 1 µmol/L)–contracted rat aortic rings with intact endothelium (except for panel d). Effect of antagonists on the contractile effect of phenylephrine (A) and the relaxing effect of factor Xa (B) in rat aortic rings. Gene induction by coagulation factor Xa is mediated by activation of protease-activated receptor 1 circres.ahajournals.org /cgi/content/full/81/5/824   (3034 words)

Factoring Cubic Functions   (Site not responding. Last check: 2007-11-06) Use this information as a resource to find more information about factoring cubic functions and related links. is an example of a cubic function with leading coefficient ?7 and constant coefficient together with forming factor rings by factoring out ideals, for quadratic functions; the techniques used to find the roots, such as factoring polynomials can closely resemble the graph of a cubic polynomial. www.flatnews.com /factoring/factoring-cubic-functions   (268 words)

Math 421 - Abstract Algebra - Course objectives Given a set with two operations, prove/disprove that it is/isn't a ring or a subring of a ring Give examples of rings that are or are not integral domains or fields Prove/disprove that a subset of a ring is/isn't an ideal www.selu.edu /Academics/Depts/Math/courses/421OBJ.htm   (84 words)

ring settings   (Site not responding. Last check: 2007-11-06) The branch of mathematics where rings are studied is called ring theory. See live article   Glossary of ring theory Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an... of contents showTocToggle("show","hide") 1 Definition of a ring 2 Types of elements 3 Homomorphisms and ideals 4 Types of rings 5 Miscellaneous Definition of a ring A ring is an abelian group (R,+)... www.byglrb.com /jewelry/ring+settings   (972 words)

Math 461 Test 1 Review Information   (Site not responding. Last check: 2007-11-06) A unit of a ring (not to be confused with the unity element of a ring) Be able to produce examples of commutative and noncommutative rings and subrings or ideals satisfying given properties. Be able to give and verify examples of ring homomorphisms similar to examples and counterexamples that we discussed in class. www.sci.uidaho.edu /m462/m462t1rev.htm   (282 words)

[No title]   (Site not responding. Last check: 2007-11-06) Elementary theory of groups, rings; polynomial rings, integral domains,.divisibility, unique factorization domains, fields, vector spaces and linear transformations. Identify and compare the properties of rings, ideals, quotient rings, integral domains, principal ideal domains, unique factorization domains, and fields.5. Understand the relationships among polynomial rings, roots of polynomials, and field extensions.TEXT:The Basics of Abstract Algebra, by Paul E. Bland, 2002 Edition, W.H. Freeman Publisher, Adopted Fall 2002. www.utm.edu /departments/math/courses/M471_Fall04.doc   (162 words)

Finite fields   (Site not responding. Last check: 2007-11-06) It is easy to tell if a quadratic or cubic is irreducible since if it were not it would have a linear factor and so the polynomial would have a root. y + 1 is a ring isomorphism from Hence the polynomial has a linear factor for each root and so cannot have more that deg(f) roots. www-groups.dcs.st-and.ac.uk /~john/MT4517/Lectures/L9.html   (909 words)

Small Rings We are being fairly arbitrary with what is a "nice" description of a ring, but these seem to fit most people's description of "nice" rings. Of particular interest are his technical report Numbers of small rings (ps-file, middle of the page) and this chart on the number of rings of prime-power order. The number of rings of size 0, 1, 2, 3, 4, 5, etc. forms the sequence 0, 1, 2, 2, 11, 2, etc., also known as sequence number A037234 from Neil Sloane's On-line Encyclopedia of Integer Sequences. home.wlu.edu /~dresdeng/smallrings   (534 words)

Factoring Polynomials Free Results   (Site not responding. Last check: 2007-11-06) fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of... Although a polynomial time algorithm exists, the most commonly used algorithm for factoring a univariate polynomial f with integer coefficients is... Factoring best suits businesses that are growing and who may... www.liquidcapitalfactoring.com /directory/factoring-polynomials-free.html   (431 words)

Math 461   (Site not responding. Last check: 2007-11-06) This course provides some of the background for students who are planning to study algebra or algebra related topics such as algebraic topology, algebraic geometry. It assumes Math 367 is known, it will start with rings, then modules. Rings, ideals, isomorphism theorems, group rings, localization, factor rings. www.math.metu.edu.tr /~semra/461   (253 words)

Ideal (ring theory) 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 concept of an order ideal that is known in order theory is derived from this notion is discussed in its dedicated article. Grow old with me! The best is yet to be. www.brainyencyclopedia.com /encyclopedia/i/id/ideal__ring_theory_.html   (1404 words)

Rings   (Site not responding. Last check: 2007-11-06) Rings is based on a game called Rubik's Rings. Although the surviving ancient rings, proved by their devices, provenance, etc... doubt, just like other people, wore rings in accordance with their station in life, for rings are... www.penelopescosmiccolors.com /ray/Rings_pcc550_67725.php   (227 words)

Review and study pointers for Exam 2 There will be no questions on 8.2 beyond the definitions of linear combinations, linear independence, basis and dimension in vector spaces. Most of the material in 6.1 was covered in the first exam, but I am including it because the concepts of homomorphism and factor rings are central to the subsequent material and should be included in any review study in preparation for the test. Although I will not ask you to reproduce the entire proof, you need to understand the process of constructing an extension field containing a zero of an irreducible polynomial by factoring out the principal ideal generated by it. www.sci.uidaho.edu /m462/m462t2rev.htm   (359 words)

ABSTRACT ALGEBRA ON LINE: Rings   (Site not responding. Last check: 2007-11-06) Definition Let R be a commutative ring with identity element 1. A ring homomorphism that is one-to-one and onto is called an isomorphism. Proposition Let I be an ideal of the commutative ring R. (a) The natural projection mapping www.math.niu.edu /~beachy/oldaaol/rings.html   (1352 words)

AMERICAN MATHEMATICAL MONTHLY -August-September 2005 We are all familiar with modular arithmetic in the integers: this is simply arithmetic in the factor ring Z It turns out that these, too, can be factored into smaller (but not always simpler) rings and this can help us gain insight into the structure of these factor rings. A connection is made to the theory of ramified primes, and a strange ring of size eight makes a surprise appearance. www.maa.org /pubs/monthly_aug_sep05_toc.html   (506 words)

Syllabus for Math 101, Joung Min Song   (Site not responding. Last check: 2007-11-06) 5.5, 5.6 Rings of Polynomials; Factorization of Polynomials over a Field 4/1 - 4/5 Review for Exam 2; Exam 2 4/8 - 4/12 Sec. 6.1, 6.2 Factor Rings and Ideals 4/15 - 4/19 Sec. There will be two midterm exams and a scheduled final exam. math.rice.edu /~jmsong/Spring02/m356/absalg.html   (556 words)

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