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Topic: Quotient ring

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	Quotient - Wikipedia, the free encyclopedia
		For example, in the problem 6 ÷ 3, the quotient would be 2, while 6 would be called the dividend, and 3 the divisor.
		A quotient can also mean just the integral part of the result of dividing two integers.
		In more abstract branches of mathematics, the word quotient is often used to describe sets, spaces, or algebraic structures whose elements are the equivalence classes of some equivalence relation on another set, space, or algebraic structure.
	en.wikipedia.org /wiki/Quotient (201 words)

	Ring (mathematics) - Wikipedia, the free encyclopedia
		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.
		Given a ring R and an ideal I of R, the quotient ring (or factor ring) R/I is the set of cosets of I together with the operations
	en.wikipedia.org /wiki/Ring_(mathematics) (1102 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.
		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.brainyencyclopedia.com /encyclopedia/i/id/ideal__ring_theory_.html (1404 words)

	Boolean ring - Wikipedia, the free encyclopedia
		A map between two Boolean rings is a ring homomorphism if and only if it is a homomorphism of the corresponding Boolean algebras.
		Furthermore, a subset of a Boolean ring is a ring ideal (prime ring ideal, maximal ring ideal) if and only if it is an order ideal (prime order ideal, maximal order ideal) of the Boolean algebra.
		The quotient ring of a Boolean ring modulo a ring ideal corresponds to the factor algebra of the corresponding Boolean algebra modulo the corresponding order ideal.
	en.wikipedia.org /wiki/Boolean_ring (507 words)

	PlanetMath: classical ring of quotients (Site not responding. Last check: 2007-10-08)
		For non-commutative rings, necessary and sufficient conditions are given by Ore's Theorem.
		Finally, note that a ring of quotients is not the same as a quotient ring.
		This is version 2 of classical ring of quotients, born on 2003-10-20, modified 2003-11-24.
	planetmath.org /encyclopedia/Regular6.html (281 words)

	Jacobson Semisimple and Quotient Rings
		The quotient map induces a group homomorphism from the units of r onto the units of r/j.
		The quotient map r/j induces an r module homomorphism from the cosets of h in r to the cosets of y in r/j.
		A simple ring has only one proper ideal, namely 0, and this must be the jacobson radical.
	www.mathreference.com /ring-jr,ss.html (734 words)

	Science Fair Projects - Polynomial ring
		In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring.
		Every commutative ring that is a finitely-generated algebra over a field can be written as a quotient of a polynomial ring.
		An interesting example of a ring obtained by using polynomials is the ring of Frobenius polynomials, where the ring multiplication is given by function composition, rather than by polynomial multiplication.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Polynomial_ring (616 words)

	[No title]
		The ideal in the free ring on the letters a, b generated by ab-ba: kernel This symbol represents a unary function.
		The kernel of a ring homomorphism is an ideal.
		The first argument is a ring R and the second argument is an element of R. When evaluated on R and such a second argument, the function represents the ideal in R generated by the second argument.
	www.win.tue.nl /~amc/oz/om/cds/ring3.html (391 words)

	Ideal (ring theory) -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		Let R be a (Jewelry consisting of a circlet of precious metal (often set with jewels) worn on the finger) ring and with (R,+) the (A group that satisfies the commutative law) abelian group of the ring.
		The ring R can be considered as a left (A self-contained component (unit or item) that is used in combination with other components) module over itself, and the left ideals of R are then seen as the (Click link for more info and facts about submodule) submodules of this module.
		The most extreme examples of factor rings are provided by (Click link for more info and facts about modding out) modding out by the most extreme ideals, and R itself.
	www.absoluteastronomy.com /encyclopedia/i/id/ideal_(ring_theory).htm (1435 words)

	PlanetMath: quotient of ideals (Site not responding. Last check: 2007-10-08)
		Some rules concerning the former type of quotient (the corresponding rules are valid also for the latter type):
		Cross-references: finitely generated, operations, ideals, intersection, Prüfer ring, quotient, inverse ideal, non-zero unity, clear, integral ideal, fractional ideals, total ring of fractions, regular elements, commutative ring
		This is version 16 of quotient of ideals, born on 2004-11-11, modified 2005-07-19.
	planetmath.org /encyclopedia/QuotientOfIdeals.html (153 words)

	Local ring - InfoSearchPoint.com (Site not responding. Last check: 2007-10-08)
		In the case of commutative rings one does not have to distinguish between left, right and two-sided ideals: a commutative ring is local if and only if it has a unique maximal ideal.
		The exact same arguments work for the ring of germs of continuous real-valued functions on any topological space at a given point, or the ring of germs of differentiable functions on any differentiable manifold at a given point, or the ring of germs of rational functions on any algebraic variety at a given point.
		The Jacobson radical m of a local ring R (which is equal to the unique left maximal ideal and also to the unique right maximal ideal) consists precisely of the non-units of the ring; furthermore it is the unique two-sided maximal ideal of R.
	www.infosearchpoint.com /display/Local_ring (1012 words)

	Quotient Rings (Site not responding. Last check: 2007-10-08)
		If the quotient ring has finite dimension (considered as a vector space over the coefficient field), further operations are available on its elements.
		Given an ideal I in the multivariate polynomial ring R (over a field), return the quotient R/I. The ideal I may either be specified as an ideal or by a list a_1, a_2,..., a_r, of generators.
		Given an element f of a finite dimensional quotient ring Q, return the representation matrix of f, which is a d by d matrix over the coefficient field of Q (where d is the dimension of Q) which represents f.
	www.math.uiuc.edu /Software/magma/text329.html (398 words)

	Ratio article - Ratio algebra quotient benzene ring carbon hydrogen naphthalene oxygen - What-Means.com (Site not responding. Last check: 2007-10-08)
		It is expressed as the quotient of one magnitude divided by another, or as a relation between several variables.
		If a school has a twenty-to-one student-teacher ratio, that means that there are twenty times as many students as teachers.
		In a benzene ring, atoms of carbon and hydrogen exist in a one-to-one ratio with each other; there are the same number of each.
	www.what-means.com /encyclopedia/Ratio (327 words)

	Quotient Rings (Site not responding. Last check: 2007-10-08)
		AMCA: Quotient rings of algebras of functions and operators by S. Jain...
		Abstract: The notion of left quotient ring, introduced by Utumi in [9], is a...
		Ideals, Principal ideals, Principal ideal rings and Quotient rings....
	www.scienceoxygen.com /math/277.html (120 words)

	blowup (Site not responding. Last check: 2007-10-08)
		T = a polynomial ring over R in variables whose weights are proportional to the degrees of the generators of J. Output values: L = the result ideal in T. The ring R may be a quotient ring.
		The ideal L defining the blowup ring R[tJ] is obtained by eliminating "t" from the ideal K produced by the script "blowup0".
		It is convenient to form the ring T by making a polynomial ring whose generators correspond to the generators of J, and then use ring-sum to tensor this ring with R. Caveats: If the ring T has more generators than needed, the correct initial subset is chosen.
	www.math.columbia.edu /online/Macaulay1-rel0994-html/node224.html (199 words)

	finite fields (Site not responding. Last check: 2007-10-08)
		The elements of this ring are 0, a, a^2, a^3,..., a^80.
		You may use ambient to see the quotient ring the field is made from.
		If you have a quotient ring that you know is a finite field, then you can convert it to ring that is known by the system to be a finite field.
	www.math.rutgers.edu /Macaulay2/0980.html (566 words)

	[No title]
		In mathematics, a Boolean ring R is a
		Furthermore, a subset of a Boolean ring is a
		quotient ring R/P is an integral domain and at the same time a Boolean ring, so it must be isomorphic to the
	en-cyclopedia.com /wiki/Boolean_ring (386 words)

	Polynomial Extensions of Finite Rings (Site not responding. Last check: 2007-10-08)
		As with all rings, the additive group is abelian.
		In this case, the additive group is isomorphic to the ring R
		The quotient ring is a field if the modulus polynomial is irreducible.
	www.hostsrv.com /webmaa/app1/MSP/webm1010/PolyExtensionRing.msp (83 words)

	Noetherian ring - InfoSearchPoint.com (Site not responding. Last check: 2007-10-08)
		In mathematics, a ring is called Noetherian if, intuitively speaking, it is not "too large" as expressed by a certain finiteness condition on its ideals.
		Noetherian rings are named after the mathematician Emmy Noether, who developed much of their theory.
		An example of a ring that's not Noetherian is a ring of polynomials in infinitely many variables: the ideal generated by these variables cannot be finitely generated.
	www.infosearchpoint.com /display/Noetherian (319 words)

	Element Operations
		Given a polynomial p over one of a certain collection of coefficient rings, this function returns a set of pairs of coefficient ring element and integer, where the ring element is a root of p in the coefficient ring, and the integer its multiplicity.
		Given elements f and g of the polynomial ring P=R[x], this function returns polynomials q (quotient) and r (remainder) in P such that f = q.g + r, and the degree of r is minimal.
		Magma calculates the polynomials q (quotient) and r (remainder) in P such that f = q * g + r, and the degree of r is minimal.
	www.math.ufl.edu /help/magma/text335.html (1370 words)

	quotient rings (Site not responding. Last check: 2007-10-08)
		For quotients of polynomial rings, a Groebner basis is computed and used to reduce ring elements to normal form after arithmetic operations.
		The presentation of the quotient ring can be obtained as a matrix with presentation.
		If a quotient ring has redundant defining relations, a new ring can be made in which these are eliminated with trim.
	www.math.uiuc.edu /Macaulay2/Manual/1442.html (251 words)

	Computer Algebra 1 -- Assignment 1
		An ideal I of the ring R is a subset of R with the properties:
		Let R be a ring and let I be an ideal in R. Define a equivalent to b mod I iff a - b is in I (a equivalent to b mod n in the integers is a special case of this).
		The elements of the quotient ring R/I are the sets I + r, where r is in R (r is called the representative of the set I + r).
	www.mcs.drexel.edu /~jjohnson/wi00/ca1/assignments/assign1.html (998 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)

	Ideals and Quotient Rings (Site not responding. Last check: 2007-10-08)
		Given a ring R and elements a_1,..., a_r of R, create the ideal I of R generated by a_1,..., a_r.
		Given a ring R and elements a_1,..., a_r of R, construct the quotient ring Q = R/I, where I is the ideal of R generated by a_1,..., a_r.
		The sum of the ideals I and J of the ring R. This ideal consists of elements a + b, with a in I and b in J. If I is generated by {a_1,..., a_k} and J is generated by {b_1,..., b_m}, then I + J is generated by {a_1,..., a_k, b_1,..., b_m}.
	www.mathematik.uni-kassel.de /magma/text342.html (438 words)

	Dense ideals and maximal quotient rings of incidence algebras.
		Our intention is to construct the maximal or Utumi ring of quotients, which is defined for any ring T, of an incidence algebra I(X, R).
		Following the alternate description given by Findlay-Lambek, we use the dense ideals of the ring to construct its maximal ring of quotients.
		Since, in general, it is hard to determine all the dense ideals of a ring, instead we construct a basis of dense ideals that form the Gabriel topology on the dense ideals.
	digitalcommons.uconn.edu /dissertations/AAI3004845 (237 words)

	Quotient Rings (Site not responding. Last check: 2007-10-08)
		Magma allows one to create the quotient ring of a multivariate polynomial ring P over a field by an ideal I of P. If the quotient ring has finite dimension (considered as a vector space over the coefficient field), further operations are available on its elements.
		Given ideals I and J of the same polynomial ring P (over a field) such that J is a ideal of I, return the quotient ring I/J. Operations on Quotient Rings
		Given a finite dimensional quotient ring Q, construct the matrix algebra A isomorphic to Q, and return A together with the isomorphism f from Q onto A. RepresentationMatrix(f) : RngQPolElt -> AlgMatElt
	www.math.uga.edu /~matthews/DOCS/MAGMA/text394.html (467 words)

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