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Ring element that can be multiplied by a nonzero element to equal 0
In abstract algebra, an element a of a ring R is called a left zero divisor if there exists a nonzero x in R such that ax = 0, or equivalently if the map
Zero_divisor
Generalizations of codimension-1 subvarieties of algebraic varieties
divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors
Divisor_(algebraic_geometry)
Graph of zero divisors of a commutative ring
in combinatorial commutative algebra, a zero-divisor graph is an undirected graph representing the zero divisors of a commutative ring. It has elements
Zero-divisor_graph
z} of a Banach algebra A {\displaystyle A} is called a topological divisor of zero if there exists a sequence x 1 , x 2 , x 3 , . . . {\displaystyle x_{1}
Topological_divisor_of_zero
Class of mathematical expression
In mathematics, division by zero, division where the divisor (denominator) is zero, is a problematic special case. Using fraction notation, the general
Division_by_zero
Esoteric, minimalist programming language
set up divisor (13) for second division loop (MEMORY LAYOUT: zero copy dividend divisor remainder quotient zero zero) >-[>+>>] Reduce divisor; Normal
Brainfuck
Largest integer that divides given integers
the greatest common divisor (GCD), also known as greatest common factor (GCF), of two or more integers, which are not all zero, is the largest positive
Greatest_common_divisor
Ring without nonzero zero divisors
the zero-product property".) Equivalently, a domain is a ring in which 0 is the only left zero divisor (or equivalently, the only right zero divisor). A
Domain_(ring_theory)
Hypercomplex number system
not a division algebra because they have zero divisors: two nonzero sedenions can be multiplied to obtain zero, for example ( e 3 + e 10 ) ( e 6 − e
Sedenion
Unique ring consisting of one element
trivial group {0}. The element 0 in the zero ring is not a zero divisor. The only ideal in the zero ring is the zero ideal {0}, which is also the unit ideal
Zero_ring
Number which when multiplied by x equals 1
nonzero element has a multiplicative inverse, but which nonetheless has divisors of zero, that is, nonzero elements x, y such that xy = 0. A square matrix has
Multiplicative_inverse
Tool in mathematical dimension theory
algebra and f a homogeneous element of degree d in A which is not a zero divisor. Then we have H S A / ( f ) ( t ) = ( 1 − t d ) H S A ( t ) . {\displaystyle
Hilbert series and Hilbert polynomial
Hilbert_series_and_Hilbert_polynomial
Number whose square is a given number
is either 0 or a zero divisor. Thus in rings where zero divisors do not exist, it is uniquely 0. However, rings with zero divisors may have multiple
Square_root
The product of two nonzero elements is nonzero
rule of zero product, the null factor law, the multiplication property of zero, the nonexistence of nonzero zero divisors, or one of the two zero-factor
Zero-product_property
Concept in mathematical ring theory
terminology by making an exception for zero divisors: one calls an element a in a commutative ring a zero divisor if there exists a nonzero x such that
Divisibility_(ring_theory)
Algebraic structure with addition and multiplication
left zero divisor of a ring R is an element a in the ring such that there exists a nonzero element b of R such that ab = 0. A right zero divisor is defined
Ring_(mathematics)
Numerous conjectures by mathematician Irving Kaplansky
torsion-free group. Kaplansky's zero divisor conjecture states: The group ring K[G] does not contain nontrivial zero divisors, that is, it is a domain. Two
Kaplansky's_conjectures
Construction within abstract algebra
commutative rings R that may have zero divisors. The construction embeds R in a larger ring, giving every non-zero-divisor of R an inverse in the larger ring
Total_ring_of_fractions
Generalizations of '"`UNIQ--math-00000046-QINU`"' in algebraic structures
additive identity among those tensors. Null semigroup Zero divisor Zero object Zero of a function Zero — non-mathematical uses Nair, M. Thamban; Singh, Arindama
Zero_element
Integer that divides another integer
In mathematics, a divisor of an integer n , {\displaystyle n,} also called a factor of n , {\displaystyle n,} is an integer m {\displaystyle m} that may
Divisor
Arithmetic operation
What is being divided is called the dividend, which is divided by the divisor, and the result is called the quotient. At an elementary level the division
Division_(mathematics)
Zero divisors in a module
torsion element is an element of a module that yields zero when multiplied by some non-zero-divisor of the ring. The torsion submodule of a module is the
Torsion_(algebra)
Module over a ring
module is a module over a ring such that zero is the only element annihilated by a regular element (non zero-divisor) of the ring. In other words, a module
Torsion-free_module
Construction of a ring of fractions
{\displaystyle s\in S,} and 0 ≠ a ∈ R {\displaystyle 0\neq a\in R} is a zero divisor with a s = 0. {\displaystyle as=0.} Then a 1 {\displaystyle {\tfrac {a}{1}}}
Localization (commutative algebra)
Localization_(commutative_algebra)
Natural number, composite number
{\displaystyle \mathbb {O} } , and holds a compact form homeomorphic to the zero divisors with entries of unit norm in the sedenions, S {\displaystyle \mathbb
14_(number)
In algebra, element without non-trivial factors
commutative rings, which is why the assumption of the ring having no nonzero zero divisors is commonly made in the definition of irreducible elements. It results
Irreducible_element
Well-behaved sequence in a commutative ring
zero-divisor on M and ri is a not a zero-divisor on M/(r1, ..., ri−1)M for i = 2, ..., d. Some authors also require that M/(r1, ..., rd)M is not zero. Intuitively
Regular_sequence
Algebraic ring that need not have additive negative elements
adjoin a new zero 0 ′ {\displaystyle 0'} to the underlying set and thus obtain such a zerosumfree semiring that also lacks zero divisors. In particular
Semiring
Algebraic structure
domain, or PID, is an integral domain (that is, a non-zero commutative ring without nonzero zero divisors) in which every ideal is principal (that is, is formed
Principal_ideal_domain
Number in {..., –2, –1, 0, 1, 2, ...}
any unital commutative ring. Certain non-zero integers map to zero in certain rings. The lack of zero divisors in the integers (last property in the table)
Integer
Set of finitely supported functions from a group to a ring
then R[G] always has zero divisors. For example, consider an element g of G of order |g| = m > 1. Then 1 − g is a zero divisor: ( 1 − g ) ( 1 + g + ⋯
Group_ring
Topics referred to by the same term
without left or right nonzero zero divisors Integral domain, a non-trivial commutative ring without nonzero zero divisors Atomic domain, an integral domain
Domain
Error-detecting code for detecting data changes
Division algorithm stops here as dividend is equal to zero. Since the leftmost divisor bit zeroed every input bit it touched, when this process ends the
Cyclic_redundancy_check
Number whose square ends in the same digits
points of f ( x ) = x 2 {\displaystyle f(x)=x^{2}} . As 0 is always a zero-divisor, 0 and 1 are always fixed points of f ( x ) = x 2 {\displaystyle f(x)=x^{2}}
Automorphic_number
Ring without non-zero nilpotent elements
+ (xy) as zero-divisors, but no non-zero nilpotent elements. As another example, the ring Z × Z contains (1, 0) and (0, 1) as zero-divisors, but contains
Reduced_ring
Four-dimensional number system
the real numbers. The next extension gives the sedenions, which have zero divisors and so cannot be a normed division algebra. The unit quaternions give
Quaternion
Number
infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted;
0
Algebraic ring without a multiplicative identity
homomorphism from a ring to a rng, and the image of f contains a non-zero-divisor of S, then S is a ring, and f is a ring homomorphism. Every rng R can
Rng_(algebra)
The Zero Divisor Theorem. If M ≠ 0 {\displaystyle M\neq 0} has finite projective dimension and r ∈ R {\displaystyle r\in R} is not a zero divisor on M
Homological conjectures in commutative algebra
Homological_conjectures_in_commutative_algebra
form of Euclid's algorithm. exact zero divisor A zero divisor x {\displaystyle x} is said to be an exact zero divisor if its annihilator, Ann R ( x )
Glossary of commutative algebra
Glossary_of_commutative_algebra
Algebraic structure designed for geometry
{1}{2}}(1+u)} is both a nontrivial idempotent element and a nonzero zero divisor, and thus has no inverse. It is usual to identify R {\displaystyle \mathbb
Geometric_algebra
Generalization of vector spaces from fields to rings
that 0 is the only element annihilated by a regular element (non zero-divisor) of the ring, equivalently rm = 0 implies r = 0 or m = 0. Noetherian A
Module_(mathematics)
Relation between genus, degree, and dimension of function spaces over surfaces
the zeros of a (non-zero) holomorphic function do not have an accumulation point. Therefore, ( f ) {\displaystyle (f)} is well-defined. Any divisor of
Riemann–Roch_theorem
is not a zero divisor then R is a complete intersection ring if and only if R/(x) is. (If the maximal ideal consists entirely of zero divisors then R is
Complete_intersection_ring
Natural number
preceding 60 (that is the composite index of 84), and 48. There are 84 zero divisors in the 16-dimensional sedenions S {\displaystyle \mathbb {S} } . 84
84_(number)
Commutative ring with no zero divisors other than zero
nonzero commutative ring with no nonzero zero divisors. An integral domain is a commutative ring in which the zero ideal {0} is a prime ideal. An integral
Integral_domain
Ideal that maps to zero a subset of a module
\{0\}}{\mathrm {Ann} _{R}(x)}.} (Here we allow zero to be a zero divisor.) In particular DR is the set of (left) zero divisors of R taking S = R and R acting on itself
Annihilator_(ring_theory)
Computation modulo a fixed integer
\mathbb {Z} /m\mathbb {Z} } , which fails to be a field because it has zero-divisors. If m > 1, ( Z / m Z ) × {\displaystyle (\mathbb {Z} /m\mathbb {Z} )^{\times
Modular_arithmetic
Smallest integer n for which n equals 0 in a ring
characteristic 1 is the zero ring, which has only a single element 0. If a nontrivial ring R does not have any nontrivial zero divisors, then its characteristic
Characteristic_(algebra)
element r of R is a called a two-sided zero divisor if it is both a left zero divisor and a right zero divisor. division A division ring or skew field
Glossary_of_ring_theory
In algebra, expression of an ideal as the intersection of ideals of a specific type
y {\displaystyle y} is in I {\displaystyle I} ; equivalently, every zero-divisor in the quotient R / I {\displaystyle R/I} is nilpotent. The radical of
Primary_decomposition
Hypercomplex number system
contain zero divisors and are thus not a division algebra. Whereas the sedenions have 84 zero divisors, the trigintaduonions have 1,260 zero divisors derived
Trigintaduonion
Abstract algebra concept
to any nonzero commutative rng R {\displaystyle R} with no nonzero zero divisors. The embedding is given by r ↦ r s s {\displaystyle r\mapsto {\frac
Field_of_fractions
Particular kind of algebraic structure
no zero divisors is isomorphic to the real or complex numbers. Every commutative real unital Noetherian Banach algebra (possibly having zero divisors) is
Banach_algebra
Algebraic structure
particular type of element is the zero divisors, i.e. an element a {\displaystyle a} such that there exists a non-zero element b {\displaystyle b} of the
Commutative_ring
Concept in commutative algebra
{\mathfrak {p}}} is prime if and only if every zero divisor in R / p {\displaystyle R/{\mathfrak {p}}} is actually zero.) Any prime ideal is primary, and moreover
Primary_ideal
Quaternions with complex number coefficients
associative, but not commutative. A biquaternion is either a unit or a zero divisor. The algebra of biquaternions forms a composition algebra and can be
Biquaternion
Result in modular arithmetic
{\mathfrak {m}}}.} Furthermore, if f ′ ( a ) {\displaystyle f'(a)} is not a zero-divisor then b is unique. This result can be generalized to several variables
Hensel's_lemma
Standard division algorithm for multi-digit numbers
problems, one number, called the dividend, is divided by another, called the divisor, producing a result called the quotient. It enables computations involving
Long_division
Construction in homological algebra
commutative ring and u {\displaystyle u} in R {\displaystyle R} is not a zero divisor, then Ext R i ( R / ( u ) , B ) ≅ { B [ u ] i = 0 B / u B i = 1 0 otherwise
Ext_functor
Algebra over a field with only invertible elements and zero
zero divisors. A finite-dimensional unital associative algebra (over any field) is a division algebra if and only if it has no nonzero zero divisors.
Division_algebra
Algebra with a graded anticommutativity property on multiplication
anticommutative algebra A over a (commutative) base ring R in which 2 is not a zero divisor is alternating. Alternating multilinear map Exterior algebra Graded-symmetric
Alternating_algebra
Four-dimensional associative algebra over the reals
split-quaternions contain nontrivial zero divisors, nilpotent elements, and idempotents. (For example, 1/2(1 + j) is an idempotent zero-divisor, and i − j is nilpotent
Split-quaternion
Topics referred to by the same term
theory, a nonzero element of a ring that is neither a left nor a right zero divisor In ring theory, a von Neumann regular element of a ring A regular element
Regular_element
other integers cannot be roots of unity modulo n, because they are zero divisors modulo n. A primitive root modulo n, is a generator of the group of
Root_of_unity_modulo_n
Analogue of a prime number in a commutative ring
Euclidean domain, or may add the additional requirement that p is not a zero-divisor. Interest in prime elements comes from the fundamental theorem of arithmetic
Prime_element
Relating two numbers and their greatest common divisor
greatest common divisor. The theorem's statement is as follows: Bézout's identity—Let a and b be integers with greatest common divisor d. Then there exist
Bézout's_identity
Type of commutative ring in mathematics
R[x] and the power series ring R[[x]] are Cohen–Macaulay. For a non-zero-divisor u in the maximal ideal of a Noetherian local ring R, R is Cohen–Macaulay
Cohen–Macaulay_ring
Method for producing composition algebras
formed by the Cayley-Dickson construction begin to have nontrivial zero divisors, in that this and every further algebra created by the construction
Cayley–Dickson_construction
relative effective Cartier divisor is roughly a family of effective Cartier divisors. Precisely, an effective Cartier divisor in a scheme X over a ring
Relative effective Cartier divisor
Relative_effective_Cartier_divisor
Fuzzy logic concept
[0, 1]. A t-norm T has zero divisors if and only if it has nilpotent elements; each nilpotent element of T is also a zero divisor of T. The set of all nilpotent
T-norm
Ideal in a ring which has properties similar to prime elements
y + 1 ) {\displaystyle (y-1)(y+1)} , which implies the existence of zero divisors in the quotient ring, preventing it from being isomorphic to C {\displaystyle
Prime_ideal
Algebra over a field where binary multiplication is not necessarily associative
also be a zero divisor. For example, all non-zero elements of the sedenions have a two-sided inverse, but some of them are also zero divisors. The free
Non-associative_algebra
Number used for counting
natural numbers a, b, and c, a × (b + c) = (a × b) + (a × c). No nonzero zero divisors: if a and b are natural numbers such that a × b = 0, then a = 0 or b
Natural_number
Element of a unital algebra over the field of real numbers
contain idempotents 1 2 ( 1 ± j ) {\textstyle {\frac {1}{2}}(1\pm j)} and zero divisors ( 1 + j ) ( 1 − j ) = 0 {\displaystyle (1+j)(1-j)=0} , so such algebras
Hypercomplex_number
Algebraic structure
greatest common divisors of a and b are associated). In particular, two polynomials that are not both zero have a unique greatest common divisor that is monic
Polynomial_ring
Concept in algebraic geometry
V {\displaystyle V} , and any divisor in it may be called a canonical divisor. An anticanonical divisor is any divisor − K {\displaystyle K} with K {\displaystyle
Canonical_bundle
Greatest common divisor of polynomials
In algebra, the greatest common divisor (frequently abbreviated GCD or gcd) of two polynomials is a polynomial, of the highest possible degree, which
Polynomial greatest common divisor
Polynomial_greatest_common_divisor
In mathematics, the theta divisor Θ is the divisor in the sense of algebraic geometry defined on an abelian variety A over the complex numbers (and principally
Theta_divisor
Concept in algebraic geometry
correspondence between line bundles and divisors (built from codimension-1 subvarieties), there is an equivalent notion of a nef divisor. More generally, a line bundle
Nef_line_bundle
Concept in algebraic geometry
between line bundles and divisors (built from codimension-1 subvarieties), there is an equivalent notion of an ample divisor. In more detail, a line bundle
Ample_line_bundle
Function of the coefficients of a polynomial that gives information on its roots
a_{n}} may not be well defined if the ring of the coefficients contains zero divisors. Such a problem may be avoided by replacing a n {\displaystyle a_{n}}
Discriminant
number projective line, and a d − b c {\displaystyle ad-bc} is not a zero divisor. A dual number is a hypercomplex number of the form x + y ε {\displaystyle
Laguerre_transformations
Abelian group in which every element can, in some sense, be divided by positive integers
rM = M for all nonzero r in R. (It is sometimes required that r is not a zero-divisor, and some authors require that R is a domain.) For every principal left
Divisible_group
total quotient ring, that is, to invert every element that is not a zero divisor. Unfortunately, in general, the total quotient ring does not produce
Function field (scheme theory)
Function_field_(scheme_theory)
Hebrew book on Jewish mysticism
"Flying Higher Than a Box-Kite: Kite-Chain Middens, Sand Mandalas, and Zero-Divisor Patterns in the 2n-ions Beyond the Sedenions". arXiv:math/0207003. Benton
Sefer_Yetzirah
would contain a non zero divisor in A / g A {\displaystyle A/gA} . However, p {\displaystyle {\mathfrak {p}}} is associated to the zero ideal in A / g A
Serre's criterion for normality
Serre's_criterion_for_normality
Study of dimension in algebraic geometry
{\displaystyle x_{1}} is not a zero-divisor on M {\displaystyle M} and x i {\displaystyle x_{i}} is not a zero divisor on M / ( x 1 , … , x i − 1 ) M
Dimension_theory_(algebra)
In mathematics, element that equals its square
Separable algebra. Any non-trivial idempotent a is a zero divisor (because ab = 0 with neither a nor b being zero, where b = 1 − a). This shows that integral domains
Idempotent_(ring_theory)
Hypercomplex number system
with the sedenions) all fail to satisfy this property. They all have zero divisors. Wider number systems exist which have a multiplicative modulus (for
Octonion
Branch of mathematics that studies algebraic structures
Jacobson radical Socle of a ring unit (ring theory), Idempotent, Nilpotent, Zero divisor Characteristic (algebra) Ring homomorphism, Algebra homomorphism Ring
List of abstract algebra topics
List_of_abstract_algebra_topics
Graph layout on multiple half-planes
from the zero divisors of a finite local ring by making a vertex for each zero divisor and an edge for each pair of values whose product is zero. In a multi-paper
Book_embedding
commutative ring. There is an algorithm for testing if an element a is a zero divisor: this amounts to solving the linear equation ax = 0. There is an algorithm
Linear_equation_over_a_ring
Period of the Fibonacci sequence modulo an integer
analysis fails for p = 2 and p is a divisor of the squarefree part of k2 + 4, since in these cases are zero divisors, so one must be careful in interpreting
Pisano_period
Number equal to the sum of its proper divisors
the sum of its positive proper divisors, that is, divisors excluding the number itself. For instance, 6 has proper divisors 1, 2, and 3, and 1 + 2 + 3 =
Perfect_number
Algorithm for computing greatest common divisors
Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers, the largest number that divides them both without
Euclidean_algorithm
Amount left over after computation
P(x) and the linear divisor x-a. Evaluate P(a), where a is the root of the linear divisor (the value that makes x-a equal to zero). The value obtained
Remainder
commutative algebra a regular ideal refers to an ideal containing a non-zero divisor. This article will use "regular element ideal" to help distinguish this
Regular_ideal
Mathematical result of division
general division). For example, when dividing 20 (the dividend) by 3 (the divisor), the quotient is 6 (with a remainder of 2) in the first sense and 6 +
Quotient
Mathematical concept
{\textstyle R_{A}} if every nonunit element of T A {\textstyle T_{A}} is a zero-divisor. Every overring of R A {\textstyle R_{A}} contained in T A {\textstyle
Overring
ZERO DIVISOR
ZERO DIVISOR
Male
African
builder; or fierce.
Boy/Male
Arabic, Australian, German, Greek, Kurdish
Empty; Void
Girl/Female
Latin Greek Shakespearean
Daughter of Priam.
Male
Italian
 Short form of Italian Raniero, NERO means "wise warrior." Compare with another form of Nero.
Male
Spanish
Spanish name derived from Latin juniperus, JUNÃPERO means "juniper tree."
Female
Greek
(ἩÏá½¼) Greek name derived form the word hÄ“rÅs, HERO means "hero." In mythology, this is the name of the lover of Leandros (Latin Leander).
Male
Croatian
, a stone.
Boy/Male
American, Australian, German, Jamaican, Latin
Strong; Vigorous; Powerful; Wise Warrior
Boy/Male
African, Finnish, German
The Lord is Exalted
Boy/Male
Australian, French, German, Greek, Italian, Portuguese
Rock; Stone
Boy/Male
Arabic
Empty.
Girl/Female
Latin
Mother of Asopus.
Boy/Male
Greek
Rock.
Girl/Female
African, Australian, French, Greek, Hebrew, Kurdish, Swahili
Seed
Male
Finnish
Finnish form of German Erich, EERO means "ever-ruler."Â
Girl/Female
Assamese, Indian
Rounded
Boy/Male
Biblical
Root, that straitens or binds, that keeps tight.
Biblical
root; that straightens or binds; that keeps tight
Male
Finnish
Short form of Finnish Antero, TERO means "man; warrior."
Biblical
crack; leak; distillation; balm
ZERO DIVISOR
ZERO DIVISOR
Girl/Female
Indian, Tamil
Devine
Boy/Male
Tamil
First, Most important, Beginning, Ornament, Adornment
Girl/Female
Muslim
Pl of Badia, Wonder, Marvel
Boy/Male
Arabic, Muslim
Brilliant; Superior; Outstanding
Girl/Female
Spanish
Boy/Male
Indian, Punjabi, Sikh
One whose Faith in God is Steadfast
Girl/Female
Australian, Czech, Czechoslovakian, German, Swedish
Battle; Female Warrior
Male
Egyptian
, Lover of Iron.
Girl/Female
American, Christian, Danish, German, Hebrew, Hindu, Indian, Swahili
Eyes; Favour; Grace
Boy/Male
Indian
A companion of the prophet (Saw)
ZERO DIVISOR
ZERO DIVISOR
ZERO DIVISOR
ZERO DIVISOR
ZERO DIVISOR
pl.
of Zero
n.
The art of calculating by nine figures and zero.
n.
The point from which the graduation of a scale, as of a thermometer, commences.
n.
A large and valuable fish of the Mackerel family, of the genus Scomberomorus. Two species are found in the West Indies and less commonly on the Atlantic coast of the United States, -- the common cero (Scomberomorus caballa), called also kingfish, and spotted, or king, cero (S. regalis).
superl.
Able; strong; valiant; redoubtable; as, a doughty hero.
n.
The common cero; also, the spotted cero. See Cero.
pl.
of Zero
n.
The principal personage in a poem, story, and the like, or the person who has the principal share in the transactions related; as Achilles in the Iliad, Ulysses in the Odyssey, and Aeneas in the Aeneid.
a.
Resembling Achilles, the hero of the Iliad; invincible.
n.
A cipher; zero.
n.
A man of distinguished valor or enterprise in danger, or fortitude in suffering; a prominent or central personage in any remarkable action or event; hence, a great or illustrious person.
n.
An illustrious man, supposed to be exalted, after death, to a place among the gods; a demigod, as Hercules.
pl.
of Hero
n.
A cipher; nothing; naught.
n.
That which has no value; a cipher; zero.
v. t.
To render worthy; to exalt into a hero.
n.
Fig.: The lowest point; the point of exhaustion; as, his patience had nearly reached zero.
n.
A Roman emperor notorius for debauchery and barbarous cruelty; hence, any profligate and cruel ruler or merciless tyrant.
n.
The character or personality of a hero.