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Zero divisor

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In abstract algebra, a non-zero element ''a'' of a ring (algebra) 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{Z}

of integers does not have any zero divisors, but in the ring

\mathbb{Z}^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{Z}/6\mathbb{Z}

, the class of 4 is a zero divisor, since 3×4 is congruent to 0 modular arithmetic modulo 6. An example of a zero divisor in the ring of 2-by-2 matrix (mathematics) 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 (mathematics) field, the left and right zero divisors coincide; they are precisely the nonzero singular matrix singular matrices. In the ring of ''n''-by-''n'' matrices over some integral domain, the zero divisors are precisely the nonzero matrices with determinant 0 (number) zero.

Properties
Left or right zero divisors can never be unit (ring theory) units, because if ''a'' is invertible and ''ab'' = 0, then 0 = ''a''⁻¹0 = ''a''⁻¹''ab'' = ''b''. Every non-zero idempotent element ''a''â‰ 1 is a zero divisor, since ''a''² = ''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 âˆž-by-âˆž 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 = \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 zero divisor and ''B'' = Transpose ''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 0â‰ 1 and without zero divisors is called an integral domain. Zero divisors occur in

\mathbb{Z}/n\mathbb{Z}

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 (mathematics) field, as every element is a unit. Zero divisors also occur in the sedenions, or 16-dimensional hypercomplex numbers under the Cayley-Dickson construction. Category:Abstract algebra Category:Ring theory de:Nullteiler es:Divisor de cero et:Nullitegur fr:Diviseur de zÃ©ro he:×ž×—×œ×§ ×?×¤×¡

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