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Topic: Trivial ring

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	PlanetMath: examples of initial objects and terminal objects and zero objects
		In the category of groups, any trivial group (consisting only of its identity element) is a zero object.
		In the category of rings with identity, the ring of integers (and any ring isomorphic to it) serves as an initial object.
		The empty scheme (which is the prime spectrum of the zero ring) is an initial object.
	planetmath.org /encyclopedia/TerminalObjectsAndZeroObjectsExamplesOfInitialObjects.html (616 words)

	Ring (mathematics) - Wikipedia, the free encyclopedia
		In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers.
		Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called commutative rings.
		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) (1183 words)

	Ring (mathematics) Information - TextSheet.com (Site not responding. Last check: 2007-10-13)
		In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers.
		In a ring we have 0=1 if and only if we are dealing with the trivial ring {0} with a single element.
		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.
	forum.top5miami.com /encyclopedia/r/ri/ring__mathematics_.html (866 words)

	Ring (mathematics) - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-10-13)
		The split-complex plane D is a ring useful in modern physics and is a subring of the tessarine s.
		For any ring R and any natural number n, the set of all square n-by-n matrices with entries from R, forms a ring with matrix addition and matrix multiplication as operations.
		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
	encyclopedia.worldsearch.com /ring_(mathematics).htm (1020 words)

	Learn more about Ring (mathematics) in the online encyclopedia. (Site not responding. Last check: 2007-10-13)
		In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers.
		Similarly, the requirement for the ring multiplication to be associative is sometimes dropped, and rings in which the associative law holds are called associative rings.
		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.onlineencyclopedia.org /r/ri/ring__mathematics_.html (835 words)

	Ring (mathematics)
		In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integer s.
		For n > 2, this matrix ring is an example of a noncommutative ring (unless R is the trivial ring).
		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 :( a+I) + ( b+I) = ( a''+''b) + I and :( a+I)
	www.nebulasearch.com /encyclopedia/article/Ring_(mathematics).html (940 words)

	Rings
		A Division Algebra is a nontrivial ring (not necessarily commutative) in which all nonzero elements are invertible.
		A homomorphism from a ring A to a ring B is a function f: A --> B such that
		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.
	mcraefamily.com /mathhelp/BasicAARings.htm (1006 words)

	Ring (mathematics) - Dangeruss-Industries.com
		In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers.
		A ring is a set R, together with two binary operations +, * acting on R such that for all a, b and c in R,
		Rings are closely related to fields which are more strictly defined.
	www.dangeruss-industries.com /Rng.html (1135 words)

	PlanetMath: matrix ring
		be a ring in the above definition is redundent.
		This is version 8 of matrix ring, born on 2006-05-08, modified 2007-01-10.
		That every ring is a matrix ring, and every matrix ring is a ring, makes the definition (though not the idea) pretty useless.
	planetmath.org /encyclopedia/MatrixRing.html (363 words)

	Trivial Magic Items
		Trivial magic items are magical devices that perform the equivalent of a cantrip spell.
		Ring of Time - Speak the command word and a monotonous voice speaks out what time of day it is in the region you're in.
		Cursed Ring of Invisibility - when it was put on the ring would become invisible, and it could only be removed with a wish spell or something similar.
	members.tripod.com /hiddenway/dnd/trivial.html (1387 words)

	Trivial ring - ExampleProblems.com
		A trivial ring is a ring defined on a singleton set, {r}.
		A ring R is trivial if and only if 1 = 0, since this equality implies that for all r within R, r = r × 1 = r × 0 = 0.
		The trivial ring is also sometimes called the zero ring, because {0} is a ring under the standard operations of addition and multiplication.
	www.exampleproblems.com /wiki/index.php/Trivial_ring (98 words)

	RING (MATHEMATICS) - GoGoSearch.com
		In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers.
		Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called commutative rings.
		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:(a I) (b I) = (a b) I and:(a I)(b I) = (ab) I.
	www.gogosearch.com /wiki/ring_(mathematics) (1365 words)

	ipedia.com: Ring (mathematics) Article (Site not responding. Last check: 2007-10-13)
		In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers.
		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
		Since any ring is both a left and right module over itself, it is possible to construct the tensor product of R over a ring S with another ring T to get another ring provided S is a central subring of R and T.
	www.ipedia.com /ring__mathematics_.html (943 words)

	FR-2.2
		A heterotricyclic parent component that consists of two benzene rings fused to a 1,4- diheterabenzene in which the heteroatoms are the same is named from the appropriate replacement prefix (Table 1 of RB-1.1 in ref 9) and the suffix '-anthrene' with elision of the 'a'.
		A heterotricyclic parent component that consists of two benzene rings fused to a 1,4- diheterabenzene in which the heteroatoms are different is named by adding the prefix 'pheno-' to the Hantzsch-Widman name (ref 9).
		Unless listed as a trivially named heterobicycle in Appendix 2 a benzene ring fused to a heteromonocycle of five or more members (a benzoheterocycle) is named by prefixing number locants indicating the position(s) of the heteroatom(s) to benzo followed by the trivial, Hantzsch-Widman, or modified replacement name (see FR-2.2.1) of the heteromonocycle without locants.
	www.chem.qmul.ac.uk /iupac/fusedring/FR22.html (1208 words)

	GAP Manual: 19 Polynomials
		The trivial polynomial and the zero of the base ring are always different.
		As was already noted in the introduction to this chapter polynomial rings are rings, so all ring functions (see chapter Rings) are applicable to polynomial rings.
		As was already noted in the introduction to this chapter Laurent polynomial rings are rings, so all ring functions (see chapter Rings) are applicable to polynomial rings.
	www.mcs.kent.edu /system/documentation/gap/CHAP019.htm (2907 words)

	Encyclopedia: Trivial ring
		A trivial ring is a ring defined on a singleton set, { x }.
		The ring operations (* and +) are trivial:
		A ring R is trivial if and only if 1 = 0, since this equality implies that for all r within R, r = r * 1 = r * 0 = 0.
	www.nationmaster.com /encyclopedia/trivial-ring (122 words)

	[No title]
		B is a map of symmetric ring spectra which is an underlying stable equivalence, then the total derived functors of restriction a* *nd ex- tension of scalars induce equivalences of homotopy theories Ho (A-mod) ~= Ho (B-mod) : Theorem 2.2.11.
		Any stable trivial cofibration is a retract of a directed colimit of pushouts* * of maps in J. Since retracts and directed colimits preserve ss-isomorphisms we only ne ed to consider pushouts of generating stable trivial* cofibrations.
		RING SPECTRA AND THH 25 The zig-zag of functors between 1 and D was investigated in section 3.
	hopf.math.purdue.edu /Shipley/whole.txt (10495 words)

	Perturbations (ring) (Site not responding. Last check: 2007-10-13)
		The properties of the ring are simple, and intuitive, transparent and already known.
		Perhaps the most essential: the fundamental group of the ring is not trivial.
		The ring is composed most of its elements, which will make or break the ring.
	www.stanford.edu /~horence/ring.html (328 words)

	Coldain Ring Quests (Site not responding. Last check: 2007-10-13)
		He then gives you the quest for the platinum ring, the details of which are still unclear.
		Once you have ring 9 and the Dirk of the Dain (result from ring 9 quest), you hand the dirk to Dain Frostreaver IV in Icewell keep and say "You can count on my help".
		Once this is done you turn in ring 9 and the head of Narandi to Seneschal Aldikar to receive the 10th coldain ring.
	www.bardthebowman.com /EQ/ColdainRingQuests.htm (2164 words)

	Articles - Characteristic (algebra) (Site not responding. Last check: 2007-10-13)
		The characteristic of the ring R may be equivalently defined as the unique natural number n such that n Z is the kernel of the unique ring homomorphism from Z to R which sends 1 to 1
		The only ring with characteristic 1 is the trivial ring which has only a single element 0=1.
		If the non-trivial ring R does not have any zero divisors, then its characteristic is either 0 or prime.
	www.kamero.net /articles/Prime_field (540 words)

	Trivial ring - Definition up Erdmond.Com (Site not responding. Last check: 2007-10-13)
		is a ring defined on a singleton_set, {x}.
		A unitary_ring R is trivial whenever 1 = 0, since this equality implies that for all r within R, r = r
		Trivial extensions of Abelian categories: Homological algebra of trivial extensions of Abelian categories with applications to ring theory (Lecture notes in mathematics ; 456)
	www.erdmond.com /Trivial_ring.html (111 words)

	Division ring - Enpsychlopedia (Site not responding. Last check: 2007-10-13)
		Stated differently, a ring is a division ring iff the group of units is the set of all non-zero elements.
		The center of a division ring is commutative and therefore a field.
		Division rings can be roughly classified according to whether or not they are finite-dimensional or infinite-dimensional over their centers.
	www.grohol.com /psypsych/Division_ring (411 words)

	Valuation Group
		Since r is a valuation ring, x/y or y/x is in r.
		If the ring is nontrivial, let x be a nonzero nonunit and note that the powers of x have increasing valuations, as described above.
		Therefore a valuation ring with a nontrivial maximal ideal cannot be artinian.
	www.mathreference.com /id-val,grp.html (673 words)

	Singular Manual: Rings and orderings
		a quotient ring by an ideal of one of 1.
		Except for quotient rings, all of these rings are realized by choosing a coefficient field, ring variables, and an appropriate global or local monomial ordering on the ring variables.
		Objects of ring dependent types are local to a ring.
	www.math.lsu.edu /singular/sing_28.htm (176 words)

	Lord of the Ring. Trivial Pursuit - Movie trilogy collectors edition - Brætspil på engelsk - Voksenspil - Spillehulen
		Classic Trivial Pursuit gameplay with 1800 Lord of the Ring questions based on the 3 movies (The Fellowship of the Ring, The Two Towers, and The Return of the King) from New Line Cinemas.
		Game features classic Trivial Pursuit gameplay for the casual Lord of the Rings movie fan but also included are the "Advanced Fan Rules".
		To win with these advanced rules a player must have 6 wedges, answer the final question in the middle and be in possession of the ring.
	www.spillehulen.dk /default.asp?Action=Details&Item=1602 (230 words)

	3S-8 and 3S9
		Fission of a ring, with addition of a hydrogen atom at each terminal group thus created, is indicated by the prefix seco-, the original steroid numbering being retained.
		It is therefore recommended that after fission of ring B the stereochemistry of substituents on ring A is indicated by the R,S convention and
		For this reason and for the benefit of biochemists, a set of trivial names has been proposed [5], which can also be used as stem names for modified compounds.
	www.chem.qmul.ac.uk /iupac/steroid/3S089.html (646 words)

	Carotenoids_Advanced.htm
		Carotenoid molecules with one or two ring structures on the ends are derived by cyclization of lycopene.
		The blue structure is a 'gamma ring' and is less common than the other two.
		There are actually three different naming systems at work, the names of the rings themselves, the 'trivial' names for the entire molecule, and the official IUPAC 'semi-systematic' names for the molecules.
	www.herbalchem.net /Carotenoids_Advanced.htm (646 words)

	Coq.ZArith.Znumtheory
		apply Zmult_one with q; auto with zarith; rewrite H0; ring.
		apply Zmult_one with (- q); auto with zarith; rewrite H0; ring.
		pattern a at 2 in - ; rewrite H2; ring*.
	pauillac.inria.fr /logical/coq/library/Coq.ZArith.Znumtheory.html (1602 words)

	Michi’s blog » Blog Archive » Localisation and ring depth
		A graded ring is - in the cases I’m interested in - always a graded algebra over some field.
		Thus, the regular sequences exhibiting lower bounds on depth are all parts of homogenous systems of parameters, but it may happen that some of the parameters occuring end up being zero-divisors, so it may happen that not all of them occur.
		The “usual” definitions tend to be for the depth of a module over a specific ring, with the more or less tacit understanding that the ring as such has depth given as the depth of the ring viewed as the obvious module over itself.
	blog.mikael.johanssons.org /archive/2006/09/localisation-and-ring-depth (1073 words)

	The One Ring \| For Online Tolkien Fans \| The Lord of the Rings Source
		So, when my Lord of the Rings geeky friends invited me to play Lord f the Rings Trivial Pursuit with them this past weekend, they were (understandably) worried that I would either dominate the game and/or go off on a few rants about the questions..and most likely, the possible answers provided by the game.
		You carry the One Ring around with you until you fail to answer a question correctly.
		Most notably, actually, was my change of heart about Frodo "offering" the Ring to the Nazgul; clearly, he was not offering the Ring: It looked to me, as pointed out by the Revisionists present, that Frodo is about to put it on his own finger.
	www.theonering.com /docs/13720.html (1356 words)

	AllaKhazam's Magical Realm - Your Everquest Community
		By gathering the drop items ahead of time (3 hours solo work), I (60 Pally) was able to do the first 5 rings in one night (2.5 hours) with a 53 Cleric buddy and two rangers in the zone (levels 51 and 52) who helped with the fights for Ring 4 and 5.
		The 6th and 7th ring could have easily been completed that same night had I done my research and been prepared (I passed up Rodrick, a rare spawn, and had to wait 3 days to eventually Multiquest his head to finish Ring#6).
		Unfortunately, I hadn't researched the 6th ring and passed up killing Rodrick who was up while I quested the first 5 (I didn't know who he was).
	everquest.allakhazam.com /db/quest.html?quest=934 (3894 words)

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