Zero divisor - Citizendia

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. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules 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 In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real 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. In Mathematics, commutativity is the ability to change the order of something without changing the end result 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 with componentwise addition and multiplication, (0,1)·(1,0) = (0,0), so both (0,1) and (1,0) are zero divisors. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French

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. In Mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in Ring theory, quite similar to the In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers

An example of a zero divisor in the ring of 2-by-2 matrices is the matrix

$\begin{pmatrix}1&1\\ 2&2\end{pmatrix}$

because for instance

$\begin{pmatrix}1&1\\ 2&2\end{pmatrix}\cdot\begin{pmatrix}1&1\\ -1&-1\end{pmatrix}=\begin{pmatrix}-2&1\\ -2&1\end{pmatrix}\cdot\begin{pmatrix}1&1\\ 2&2\end{pmatrix}=\begin{pmatrix}0&0\\ 0&0\end{pmatrix}$

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 Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Linear algebra, an n -by- n (square matrix A is called invertible or non-singular if there exists an n -by- In the ring of n-by-n matrices over some integral domain, the zero divisors are precisely the nonzero matrices with determinant zero. 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 In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n

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, a₃. . . ). 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

$A = \begin{pmatrix} 0 & 1 & 0 &0&0&\\ 0 & 0 & 1 &0&0&\cdots\\ 0 & 0 & 0 &1&0&\\ 0&0&0&0&1&\\ &&\vdots&&&\ddots \end{pmatrix}$

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. This article is about the Matrix Transpose operator For other uses see Transposition In Linear algebra, the transpose of a That AB is the identity matrix is the same as saying LR is the identity. In Linear algebra, the identity matrix or unit matrix of size n is the n -by- n Square matrix with ones on the Main 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. In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i

Every nonzero idempotent element a ≠ 1 is a zero divisor, since a² = a implies a(a − 1) = (a − 1)a = 0. Idempotence ˌaɪdɨmˈpoʊtəns describes the property of operations in Mathematics and Computer science which means that multiple applications of the operation Nonzero nilpotent ring elements are also trivially zero divisors. In Mathematics, an element x of a ring R is called nilpotent if there exists some positive Integer n such that

A commutative ring with 0 ≠ 1 and without zero divisors is called an integral domain. In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property 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

Zero divisors occur in the quotient ring Z/nZ if and only if n is composite. In Mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in Ring theory, quite similar to the A composite number is a positive Integer which has a positive Divisor other than one or itself When n is prime, there are no zero divisors and this ring is, in fact, a field, as every element is a unit. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i

Zero divisors also occur in the sedenions, or 16-dimensional hypercomplex numbers under the Cayley-Dickson construction. In Abstract algebra, sedenions form a 16- dimensional algebra over the reals. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it The term hypercomplex number has been used in Mathematics for the elements of algebras that extend or go beyond Complex number arithmetic

Set of zero divisors is a union of prime ideals. In Mathematics, a prime ideal is a Subset of a ring which shares many important properties of a Prime number in the Ring of integers

Examples

Properties

See also