Zero divisor

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In abstract algebra, a nonzero element a of a ring A is a left divisor of zero if there exists a nonzero element null b such that ab = 0. The right divisors of zero are defined analogously. An element that is both a left and right divisor of zero is called a zero divisor. If the product is commutative, then there is no need to distinguish between left and right divisors of zero. A non-null element that is not a left or right divisor of zero is called regular.

Definition

Let a≠ 0 and b ≠ 0 be two distinct elements of a ring R such that ab = 0. a and b are called zero divisors, if a is left divisor and b is right divisor.

Examples

The ring Z of the integers has no divisors of zero, but in the ring Z × Zor Z² (where the sum and the product are made component to component), it has to (0.1) × (1.0) = (0.0), so both (0.1) and (1.0) are divisors of zero.

In the quotient ring Z/6Z, class 4 is a zero divider, as 3×4 is congruent with 0 module 6. In general, zero dividers exist in the ring Z/nZ Yes and only if n is composite number and correspond to those numbers that are not relative cousins with n.

An example of zero divider on the left in the 2×2 matrice ring is the following matrix:

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

because, for example,

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

, and the other matrix is from zero to right

Sean f and g two actual functions, none is the function zero, real variable defined by

{displaystyle f(x)=0,xleq 0}

and

0}" xmlns="http://www.w3.org/1998/Math/MathML">f(x)=x,x▪0{displaystyle f(x)=x,x0}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c3ca8f41d518a091e31b2fbf316323e2613f011" style="vertical-align: -0.838ex; width:15.47ex; height:2.843ex;"/>

{displaystyle g(x)=x,xleq 0}

and

0}" xmlns="http://www.w3.org/1998/Math/MathML">g(x)=0,x▪0{displaystyle g(x)=0,x0}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10b7d480a549d7e983878a7a8d5ec21ea0500aa8" style="vertical-align: -0.838ex; width:15.14ex; height:2.843ex;"/>

The product

{displaystyle fcdot g=0}

for everything xReal number.

Counterexamples

When p is a prime number, the ring Z_p has no divisors of zero. Since every element of the ring is a unit, the ring is a body.

Properties

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

Every non-zero idempotent element a≠1 is a divisor of zero, since a² = a implies that a(a - 1) = (a - 1)a = 0. Nilpotent elements do not nulls of the ring are also trivial divisors of zero.

In the ring of n×n matrices over some field, the left and right divisors of zero coincide; are precisely the nonzero singular matrices. In the ring of n×n matrices over an integrity domain, the divisors of zero are precisely the nonzero matrices of zero determinant.

If a is a left divisor of zero and x is an arbitrary element of the ring, then xa is either zero or a zero divisor. The following example shows that the same cannot be said of ax. Consider the set of ∞×∞ matrices over the ring of integers, where each row and each column contains a finite number of nonzero entries. This is a ring with the usual matrix product. Matrix

{displaystyle 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}}}

is a left divisor of zero and B = AT is therefore a right divisor of zero. But AB is the identity matrix and therefore cannot be a divisor of zero. In particular, we conclude that A cannot be a right divisor of zero.

A commutative ring with 0≠1 and no divisors of zero is called an integrity domain or integral domain.

The cancellation laws are valid in a ring R if and only if R has no divisors of 0. In this case the equation ax = b has a unique solution.

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