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Encyclopedia > 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 b such that ba = 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. Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In mathematics, an element (also called a member) is an object contained in a set (or more generally a class). ... 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. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...


Examples

  • The ring Z of integers has no zero divisors, but in the ring Z × Z where addition and multiplication are carried out componentise 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
begin{pmatrix}1&1 2&2end{pmatrix}

because for instance The integers are commonly denoted by the above symbol. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ...

begin{pmatrix}1&1 2&2end{pmatrix}cdotbegin{pmatrix}1&1 -1&-1end{pmatrix}=begin{pmatrix}-2&1 -2&1end{pmatrix}cdotbegin{pmatrix}1&1 2&2end{pmatrix}=begin{pmatrix}0&0 0&0end{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 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,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. 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 same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ... In abstract algebra, 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; that is, there are no zero divisors. ... 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 is regarded as a linear transformation. ... For other senses of this word, see zero or 0. ...


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 = AT 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. In linear algebra, the transpose of a matrix A is another matrix AT (also written Atr, tA, or A′) created by any one of the following equivalent actions: write the rows of A as the columns of AT write the columns of A as the rows of AT reflect A... 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. ...


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 ring R is an element u such that there is v in R with uv = vu = 1R. That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of...


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. In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ... In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0. ...



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 obeys the commutative law. ... In abstract algebra, 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; that is, there are no zero divisors. ...


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. In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... A composite number is a positive integer which has a positive divisor other than one or itself. ... In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... 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 same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a unit in a ring R is an element u such that there is v in R with uv = vu = 1R. That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of...


Zero divisors also occur in the sedenions, or 16-dimensional hypercomplex numbers under the Cayley-Dickson construction. The sedenions form a 16-dimensional algebra over the reals obtained by applying the Cayley-Dickson construction to the octonions. ... 2-dimensional renderings (ie. ... The term hypercomplex number has been used in mathematics for the elements of algebras that extend or go beyond complex number arithmetic. ... 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. ...


See also


  Results from FactBites:
 
Zero divisor - Wikipedia, the free encyclopedia (446 words)
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.
  More results at FactBites »

 
 

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