Zero divisors

In abstract algebra, a nonzero element of a ring is called a left zero divisor if there exists a nonzero such that ab = 0, similarly a nonzero element of a ring is called a right zero divisor if there exists a nonzero such that ba = 0.[1] This is a partial case of divisibility in rings. An element that is a left or a right zero divisor is simply called a zero divisor.[2] An element  that is both a left and a right zero divisor[3] is called a two-sided zero divisor. If the ring is commutative, then the left and right zero divisors are the same. A non-zero element of a ring that is not a zero divisor is called regular.

Examples

2&2\end{pmatrix}, because for instance \begin{pmatrix}1&1\\ 2&2\end{pmatrix}\begin{pmatrix}1&1\\ -1&-1\end{pmatrix}=\begin{pmatrix}-2&1\\ -2&1\end{pmatrix}\begin{pmatrix}1&1\\ 2&2\end{pmatrix}=\begin{pmatrix}0&0\\ 0&0\end{pmatrix}.

  • Actually, the simplest example of a pair of zero divisor matrices is

\begin{pmatrix}1&0\\0&0\end{pmatrix} \begin{pmatrix}0&0\\0&1\end{pmatrix} = \begin{pmatrix}0&0\\0&0\end{pmatrix} = \begin{pmatrix}0&0\\0&1\end{pmatrix} \begin{pmatrix}1&0\\0&0\end{pmatrix}.

  • A direct product of two or more non-trivial rings always has zero divisors similarly to the 2\times 2-matrix example just above (the ring of diagonal 2\times 2 matrices over a ring R is the same as the direct product R\times R).

One-sided zero-divisor

  • Consider the ring of (formal) matrices \begin{pmatrix}x&y\\0&z\end{pmatrix} with x,z\in\mathbb{Z} and y\in\mathbb{Z}/2\mathbb{Z}. Then \begin{pmatrix}x&y\\0&z\end{pmatrix}\begin{pmatrix}a&b\\0&c\end{pmatrix}=\begin{pmatrix}xa&xb+yc\\0&zc\end{pmatrix} and \begin{pmatrix}a&b\\0&c\end{pmatrix}\begin{pmatrix}x&y\\0&z\end{pmatrix}=\begin{pmatrix}xa&ya+zb\\0&zc\end{pmatrix}. If x\ne0\ne y, then \begin{pmatrix}x&y\\0&z\end{pmatrix} is a left zero divisor iff x is even, since \begin{pmatrix}x&y\\0&z\end{pmatrix}\begin{pmatrix}0&1\\0&0\end{pmatrix}=\begin{pmatrix}0&x\\0&0\end{pmatrix}; and it is a right zero divisor iff z is even for similar reasons. If either of x,z is 0, then it is a two-sided zero-divisor.
  • Here is another 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 \mathrm{End}(S), the endomorphism ring 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,a4,...), and the projection map onto the first factor P(a1,a2,a3,...)=(a1,0,0,...). All three of these additive maps are not zero, and the composites LP and PR 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. Note also that RL is a two-sided zero-divisor since RLP=0=PRL, while LR=1 is not in any direction.

Non-examples

The ring of integers modulo a prime number does not have zero divisors and this ring is, in fact, a field, as every non-zero element is a unit.

More generally, there are no zero divisors in division rings.

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

Properties

In the ring of -by- matrices over some field, the left and right zero divisors coincide; they are precisely the non-zero singular matrices. In the ring of -by- matrices over some integral domain, the zero divisors are precisely the non-zero matrices with determinant zero.

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

Every non-trivial idempotent element in a ring is a zero divisor, since a2 = a implies that a  (a − 1) = a2a = aa = 0, with nontriviality ensuring that neither factor is 0. Nonzero nilpotent ring elements are also trivially zero divisors.

The set of zero divisors is the union of the associated prime ideals of the ring.

See also

Notes

References

  • MathWorld.
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