Information about Zero Divisor

In abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0. Right zero divisors are defined analogously, that is, a nonzero element a of a ring is a right zero divisor if there exists a nonzero c such that ca = 0. An element that is both a left and a right zero divisor is simply called a zero divisor. If multiplication in the ring is commutative, then the left and right zero divisors are the same. A nonzero element of a ring that is neither left nor right zero divisor is called regular.

Examples

• The ring Z of integers has no zero divisors, but in the ring Z × Z where addition and multiplication are carried out componentwise we have (0,1) × (1,0) = (0,0) and so both (0,1) and (1,0) are zero divisors.
• In the factor ring Z/6Z, the class of 4, or 4 + 6Z, is a zero divisor, since 3 × 4 is congruent to 0 modulo 6.
• An example of a zero divisor in the ring of 2-by-2 matrices is the matrix

because for instance

• More generally in the ring of n-by-n matrices over some field, the left and right zero divisors coincide; they are precisely the nonzero singular matrices. In the ring of n-by-n matrices over some integral domain, the zero divisors are precisely the nonzero matrices with determinant zero.
• Here is an example of a ring with an element that is a zero divisor on one side only. Let S be the set of all sequences of integers (a₁, a₂,a_{3...). Take for the ring all additive maps from S to S, with pointwise addition and composition as the ring operations. (That is, our ring is End(S), the endomorphisms of the additive group S.) Three examples of elements of this ring are the right shift R(a1, a2,a3,...) = (0, a1, a2,...), the left shift L(a1, a2,a3,... ) = (a2, a3,...), and a third additive map T(a1, a2,a3,... ) = (a1, 0, 0, ... ). All three of these additive maps are not zero, and the composites LT and TR are both zero, so L is a left zero divisor and R is a right zero divisor in the ring of additive maps from S to S. However, L is not a right zero divisor and R is not a left zero divisor: the composite LR is the identity, so if some additive map f from S to S satisfies fL= 0 then composing both sides of this equation on the right with R shows (fL)R = f(LR) = f1 = f has to be 0, and similarly if some f satisfies Rf = 0 then composing both sides on the left with L shows f is 0.}

Continuing with this example, note that while RL is a left zero divisor ((RL)T = R(LT) is 0 because LT is), LR is not a zero divisor on either side because it is the identity.

Concretely, we can interpret additive maps from S to S as countably infinite matrices. The matrix

realizes L explicitly (just apply the matrix to a vector and see the effect is exactly a left shift) and the transpose B = A^T realizes the right shift on S. That AB is the identity matrix is the same as saying LR is the identity. In particular, as matrices A is a left zero divisor but not a right zero divisor.

Properties

Left or right zero divisors can never be units, because if a is invertible and ab = 0, then 0 = a⁻¹0 = a⁻¹ab = b.

Every nonzero idempotent element a ≠ 1 is a zero divisor, since a² = a implies a(a − 1) = (a − 1)a = 0. Nonzero nilpotent ring elements are also trivially zero divisors.

A commutative ring with 0 ≠ 1 and without zero divisors is called an integral domain.

Zero divisors occur in the quotient ring Z/nZ if and only if n is composite. When n is prime, there are no zero divisors and this 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.

See also

• Integral domain
• Zero-product property

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. Most authors nowadays simply write algebra instead of abstract algebra.
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In mathematics, the elements or members of a set (or more generally a class) are all those objects which when collected together make up the set (or class).

Set theory and elements

Writing , means that the elements of the set are the numbers 1, 2, 3 and 4.
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In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. The branch of abstract algebra which studies rings is called ring theory.
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Commutativity is a widely used mathematical term that refers to the ability to change the order of something without changing the end result. It is a fundamental property in most branches of mathematics and many proofs depend on it.
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The integers (from the Latin integer, which means with untouched integrity, whole, entire) are the set of numbers including the whole numbers (0, 1, 2, 3, …) and their negatives (0, −1, −2, −3, …).
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In mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in ring theory, quite similar to the factor groups of group theory and the quotient spaces of linear algebra.
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Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value — the modulus.
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matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied.
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field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers.
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invertible or non-singular if there exists an n-by-n matrix such that

where denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication.
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In abstract algebra, a branch of mathematics, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠ 1, in which the product of any two non-zero elements is always non-zero (the zero-product property); that is, there
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In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A
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0 (zero) is both a number and a numerical digit used to represent that number in numerals. It plays a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures.
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In linear algebra, the transpose of a matrix A is another matrix A^T (also written A^tr, ^tA, or A′) created by any one of the following equivalent actions:

• write the rows of A

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In linear algebra, the identity matrix or unit matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere.
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In mathematics, a unit in a (unital) ring R is an invertible element of R, i.e. an element u such that there is a v in R with

uv = vu = 1_R, where 1_R

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Idempotence IPA: /ˌaɪdɨmˈpoʊtənts/ describes the property of operations in mathematics and computer science that yield the same result after the operation is applied multiple times.
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nilpotent if there exists some positive integer n such that xⁿ = 0.

Examples

• This definition can be applied in particular to square matrices.

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In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. This means that if a and b are any elements of the ring, then a×b=b×a.
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In abstract algebra, a branch of mathematics, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠ 1, in which the product of any two non-zero elements is always non-zero (the zero-product property); that is, there
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In mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in ring theory, quite similar to the factor groups of group theory and the quotient spaces of linear algebra.
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A composite number is a positive integer which has a positive divisor other than one or itself. By definition, every integer greater than one is either a prime number or a composite number. The number one is considered to be neither prime nor composite.
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In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. An infinitude of prime numbers exists, as demonstrated by Euclid in about 300 BC.
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In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. The branch of abstract algebra which studies rings is called ring theory.
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field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers.
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In mathematics, a unit in a (unital) ring R is an invertible element of R, i.e. an element u such that there is a v in R with

uv = vu = 1_R, where 1_R

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Sedenions form a 16-dimensional algebra over the reals. The set of sedenions is denoted as . Two types are currently known:

1. Sedenions obtained by applying the Cayley-Dickson construction

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dimension (Latin, "measured out") is a parameter or measurement required to define the characteristics of an object—i.e., length, width, and height or size and shape.
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The term hypercomplex number has been used in mathematics for the elements of algebras that extend or go beyond complex number arithmetic. Hypercomplex numbers have had a long lineage of devotees including Hermann Hankel, Georg Frobenius, Eduard Study, and Elie Cartan.
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In mathematics, the Cayley-Dickson construction produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. The algebras produced by this process are known as Cayley-Dickson algebras
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