Zero divisor

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In abstract algebra, a non-zero element a of a ring R is a left zero divisor if there exists a non-zero b such that ab = 0. Right zero divisors are defined analogously, that is, a non-zero element a of a ring R is a right zero divisor if there exists a non-zero b such that ba = 0. An element that is both a left and a right zero divisor is simply called a zero divisor. If the multiplication is commutative, then one does not have to distinguish between left and right zero divisors. A non-zero element that is neither left nor right zero divisor is called regular.

Examples

The ring [\mathbb] of integers does not have any zero divisors, but in the ring [\mathbb^2] (where addition and multiplication are carried out component wise), we have (0,1) × (1,0) = (0,0) and so both (0,1) and (1,0) are zero divisors.

In the factor ring [\mathbb/6\mathbb], the class of 4 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

[\begin1&1\\2&2\end]

because for instance

[\begin1&1\\2&2\end\cdot\begin1&1\\-1&-1\end=\begin-2&1\\-2&1\end\cdot\begin1&1\\2&2\end=\begin0&0\\0&0\end]

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.

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

If a is a left zero divisor, and x is an arbitrary ring element, then xa is either zero or a left zero divisor. The following example shows that the same cannot be said about ax. Consider the set of ∞-by-∞ matrices over the ring of integers, where every row and every column contains only finitely many non-zero entries. This is a ring with ordinary matrix multiplication. The matrix

[A = \begin0 & 1 & 0 &0&0&\\0 & 0 & 1 &0&0&\cdots\\0 & 0 & 0 &1&0&\\0&0&0&0&1&\\&&\vdots&&&\ddots\end]

is a left zero divisor and B = A^T is therefore a right zero divisor. But AB is the identity matrix and hence certainly not a zero divisor. In particular, we can conclude that A cannot be a right zero divisor.

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

Zero divisors occur in [\mathbb/n\mathbb] if and only if n is composite. 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.

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