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Topic: Zero divisors

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	Zero divisor - Wikipedia, the free encyclopedia
		In the factor ring Z/6Z, the class of 4 is a zero divisor, since 3×4 is congruent to 0 modulo 6.
		When n is prime, there are no zero divisors and this factor ring is, in fact, a field, as every element is a unit.
		Zero divisors also occur in the sedenions, or 16- dimensional hypercomplex numbers under the Cayley-Dickson construction.
	en.wikipedia.org /wiki/Zero_divisor (446 words)

	Division by zero Info - Bored Net - Boredom (Site not responding. Last check: 2007-10-07)
		It is possible to disguise a division by zero in a long algebraic argument, leading to such things as a spurious proof that 2 equals 1.
		Although division by zero is undefined with real numbers and integers, it is possible to consistently define division by zero in other number systems.
		Zero divisors are frequently found in group theory and in hyperreal numbers and surreal numbers.
	www.borednet.com /e/n/encyclopedia/d/di/division_by_zero.html (316 words)

	Zero divisor -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-07)
		But AB is the (A scalar matrix in which all of the diagonal elements are unity) identity matrix and hence certainly not a zero divisor.
		When n is prime, there are no zero divisors and this factor ring is, in fact, a (A piece of land cleared of trees and usually enclosed) field, as every element is a unit.
		Zero divisors also occur in the (Click link for more info and facts about sedenions) sedenions, or 16- dimensional (Click link for more info and facts about hypercomplex numbers) hypercomplex numbers under the (Click link for more info and facts about Cayley-Dickson construction) Cayley-Dickson construction.
	www.absoluteastronomy.com /encyclopedia/z/ze/zero_divisor.htm (650 words)

	Encyclopedia: Zero divisor
		An example of a zero divisor in the ring of 2-by-2 matrices is the matrix For the square matrix section, see square matrix.
		In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers.
		In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere.
	www.nationmaster.com /encyclopedia/Zero-divisor (1087 words)

	Homework
		So a good answer is: The zero divisors are 0 and all numbers less than or equal 76 which are multiples of 7 or of 11 (or of both).
		The zero divisors mod 7·11·13 = 1001 are 0 and all numbers up to 1000 which are multiples of 7 or of 11 or of 13.
		The non-zero zero divisors mod 12 are: 2,3,4, 6, 8, 9, 10.
	www.math.fau.edu /HTMLFILE/Outlines/mfla2f00h9s.html (1097 words)

	Integral domain - Wikipedia, the free encyclopedia
		Alternatively and equivalently, integral domains may be defined as commutative rings in which the zero ideal {0} is prime, or as the subrings of fields.
		Viewing the underlying commutative ring as a categorical construction, the previous criterion on zero divisors is equivalent to the condition that every nonzero morphism is a monomorphism (hence also an epimorphism).
		If a and b are elements of the integral domain R, we say that a divides b or a is a divisor of b or b is a multiple of a if and only if there exists an element x in R such that ax = b.
	en.wikipedia.org /wiki/Integral_domain (714 words)

	Integral Domains
		We found that neither Z nor Z[ m], where m is not a complete square of an integer, have no divisors of zero.
		, 2 and 3 are divisors of zero and, therefore, are not invertible.
		The remaining non-zero element 4 is a divisor of zero.
	www.cut-the-knot.com /arithmetic/summary.shtml (403 words)

	[No title]
		Usually the zero divisors are present in different algebraic contexts 2 1) the trivial one: in the direct sum of algebras i.e.
		In Chapter II we construct some type of zero divisors (called special ones), in A 4are all of them, which are pairs of alternatives elements, we describe all of them completely for n 4 and relate these zero divisors with the multiplicative monomorphisms of A 3to A n n 3.
		A zero divisor (a; b) in A n+1 is special if (a; b)(x; y) = (0; 0) implies that {a; y; b} is one special triple.
	hopf.math.purdue.edu /Moreno/cdivisor.txt (4729 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-07)
		It just seems so odd to me that zero, the very center of our entire concept of mathematics (something long held to be the purest, most perfect thing in existence) should have such gaping flaws.
		After all, that is how we got from natural numbers to integers (appending negative integers and zero), and from integers to rationals (appending ratios of integers), and from rationals to reals (appending limits of convergent sequences), and from reals to complexes (appending the square root of -1).
		Zero just has the property that you can't divide by it.
	www.mathforum.org /library/drmath/view/55764.html (861 words)

	PlanetMath: zero polynomial (Site not responding. Last check: 2007-10-07)
		All coefficients of the zero polynomial are equal to 0, i.e.
		Cross-references: zero divisors, polynomial ring, identity element, polynomials, ring
		This is version 7 of zero polynomial, born on 2004-10-29, modified 2005-04-27.
	planetmath.org /encyclopedia/ZeroPolynomial2.html (101 words)

	Mathematics for Liberal Arts II
		Counting numbers having inverses is similar to counting zero divisors, a number is either one or the other.
		11, the zero divisors are all numbers divisible by 5 or by 11.
		One zero divisor is 0; then we have 10 non-zero multiples of 5 and 4 non-zero multiples of 11; a total of 1 + 10 + 4 = 15 zero divisors.
	www.math.fau.edu /schonbek/mfla/mfla2sp03se3s.html (1711 words)

	Math Forum - Ask Dr. Math
		Date: 10/06/97 at 12:47:29 From: Doctor Rob Subject: Re: The number 1 and zero One is neither a prime nor a composite number.
		Zero is not a prime or a composite number either.
		Zero has an infinite number of divisors (any nonzero whole number divides zero).
	www.forum.swarthmore.edu /dr.math/problems/christy10.6.97.html (317 words)

	PlanetMath: zero divisor (Site not responding. Last check: 2007-10-07)
		is said to be a zero divisor if it is both a left and right zero divisor.
		is said to be a regular element if it is neither a left nor a right zero divisor.
		This is version 5 of zero divisor, born on 2002-07-06, modified 2005-04-20.
	planetmath.org /encyclopedia/ZeroDivisor.html (82 words)

	[No title] (Site not responding. Last check: 2007-10-07)
		But this proof is invalid in $\zmodnz{15}$, or in any other ring with zero divisors, because it is not possible to cancel.
		That is to say, consider a zero divisor $z$ in $\bigoplus_{i = 1}^{r} \zmodnz { p_i^{\nu_i}}$.
		The $\zmodnz 5$ component is ``squeezed" to zero by $\lambda$, and the localized ring is isomorphic to $\zmodnz 3$.
	math.isu.edu /~wolperj/local.text (4212 words)

	Divisors And Euler Sparks Of Atomic Sections (ResearchIndex) (Site not responding. Last check: 2007-10-07)
		Divisors And Euler Sparks Of Atomic Sections (ResearchIndex)
		The first purpose is to give a simple treatment of the basic theory of atomic sections and their zero currents or divisors.
		8 Zero divisors of atomic functions (context) - Harvey, Semmes - 1992
	citeseer.ist.psu.edu /197854.html (331 words)

	Division by a High Matrix (Site not responding. Last check: 2007-10-07)
		Note that, since none of the diagonal entries of the diagonal matrix D are zero, DZ determines Z. Therefore, the solution method enables you to recover the original X when A and Y = AX are given.
		Typically, a high matrix, like a square matrix, will not be a left zero divisor, so when A is high it will normally be possible to recover X when X and AX are known.
		Note that there is no really good disguise in the 3x2 case, but there are examples of 4x3 left zero divisors that are not immediately recognizable as such.
	www.nyu.edu /classes/edwards/chap3/high.html (290 words)

	The Ultimate Talk:Zero divisor Dog Breeds Information Guide and Reference (Site not responding. Last check: 2007-10-07)
		The given infinite matrix A is a left zero divisor, as A * (1and0and0and...\\ 0and0and0...\\...) = 0, but it is not a right zero divisor, because C*A=0 implies C=0.
		In contrast to that, B is not a left zero divisor, but a right zero divisor (using the same matrix as for A as the left factor).
		Also it is only proven there that a left zero divisor cannot have a left inverse (a'a=1), but it may still have a right inverse (aa'=1).
	www.dogluvers.com /dog_breeds/Talk:Zero_divisor (187 words)

	Using the GNU Compiler Collection (GCC): M88K Options (Site not responding. Last check: 2007-10-07)
		Do, or don't, generate code to guarantee that integer division by zero will be detected.
		By default, when compiling code that might be run on such a processor, GCC generates code that explicitly checks for zero-valued divisors and traps with exception number 503 when one is detected.
		GCC assumes that the MC88110 processor correctly detects all instances of integer division by zero.
	www.ifh.ee.ethz.ch /sepp/gcc-3.2.2-to/gcc_31.html (936 words)

	ZeroDivisor Algebras, Charles Muses
		Charles Muses has described some algebras, containing Zero Divisors, which he considers to be a more fundamental concept than Division Algebras.
		Robert de Marrais has constructed detailed models for Zero Divisors of algebras of dimension 2^N, with explicit results through N = 8.
		Guillermo Moreno has shown that the zero divisors of the sedenions S whose basis entries are norm one are homeomorphic to the 14-dimensional Lie group G2 = Aut(O).
	www.valdostamuseum.org /hamsmith/NDalg.html (3990 words)

	Zero divisor : Zero divisors (Site not responding. Last check: 2007-10-07)
		terms defined : Zero divisor : Zero divisors
		All is still licensed under the GNU FDL.
		I, therefore, when I hear thy voice, do say, for He will speak.html">speak comfort; He saith, He hath got the victory, His most glorious conquest.
	www.termsdefined.net /ze/zero-divisors.html (95 words)

	Welcome to Mathsoft
		The left zero divisors constitute a finite union of subspaces for each of the six representative R-algebras.
		So for a fixed random 2-associative algebra A, one can readily sample four independent zero divisors at a time and determine whether they are coplanar.
		Observe that the zero divisor structure proof for 3-dimensional, 3-associative algebras with identity rested on a classification theorem.
	www.mathsoft.com /mathsoft_resources/unsolved_problems/2190.asp (916 words)

	Computer Algebra 1 -- Assignment 1
		The cancellation property is equivalent to the ring not having any zero divisors.
		A prime is an element p such that the only divisors of p are p itself and units.
		Hint, show that for any a, not equal to zero, in Z/p, that (a), the ideal generated by a is equal to Z/p.
	www.mcs.drexel.edu /~jjohnson/wi00/ca1/assignments/assign1.html (998 words)

	13 (Site not responding. Last check: 2007-10-07)
		The numbers that are in the first list for each type of product are called divisors of zero.
		Find all of the divisors of zero for products modulo 9, products modulo 10, products modulo 11 and products modulo 12.
		When using multiplication modulo n, some numbers are not divisors of zero.
	home.snu.edu /~lturner.fs/MC-MathStr/GCDInClass3.htm (283 words)

	NSDL Metadata Record -- zero divisor
		The element a is a left zero divisor if there exists a nonzero element...
		The element a is said to be a zero divisor if it is both a left and right zero divisor.
		Then the elements 2 and 3 are zero divisors, since...
	nsdl.org /mr/1034953 (90 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-07)
		A zero-divisor in a ring is an element of the ring which, when multiplied by some non-zero ring element, gives a product of zero.
		Many other examples of rings are known, some with nonzero zero- divisors and some without.
		Their study is one of the areas of modern abstract algebra which is covered in upper-level undergraduate and lower-level graduate courses.
	mathforum.org /library/drmath/view/52410.html (581 words)

	WAM Abstracts
		For example, a nilpotent ring is radical and consists entirely of zero divisors.
		We consider the algebraic implications that certain properties have on zero divisors in Jacobson radical rings and explore related properties such as topological nilpotents in Banach algebras.
		It is known that J(M)= M, and the absence of nonzero divisors of zero in M is equivalent to the Theorem of Titchmarsh from analysis.
	www.math.fau.edu /weekend-algebra/abstracts.html (2230 words)

	Ax::noZeroDivisors -- the axiom of rings with no zero divisor (Site not responding. Last check: 2007-10-07)
		states that a ring without a unit has no zero divisors.
		is used to state that a ring without a unit has no zero divisors, i.e., that the product of two non-zero elements is never zero.
		Note that an integral domain implicitly has no zero divisors.
	www.mupad.de /doc/25/de/Ax/noZeroDivisors.shtml (74 words)

	Modular Arithmetic, Differences and Similarities
		In usual arithmetic, ab = 0 is only possible when either a or b is zero.
		Arithmetic modulo a prime, too, has no zero divisors.
		Fundamental Theorem of Algebra implies that every polynomial of order n over the field C of complex numbers has exactly n roots.
	www.cut-the-knot.com /blue/sim_diff.shtml (374 words)

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