Zero divisor

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In abstract algebra, an element a of a ring R is called a left zero divisor if there exists a nonzero x such that ax = 0, or equivalently if the map from R to R that sends x to ax is not injective.[1] Similarly, an element a of a ring is called a right zero divisor if there exists a nonzero y such that ya = 0. 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 a that is both a left and a right zero divisor is called a two-sided zero divisor (the nonzero x such that ax = 0 may be different from the nonzero y such that ya = 0). If the ring is commutative, then the left and right zero divisors are the same.

An element of a ring that is not a zero divisor is called regular, or a non-zero-divisor. A zero divisor that is nonzero is called a nonzero zero divisor or a nontrivial zero divisor.

Examples[edit]

  • In the ring \mathbb{Z}/4\mathbb{Z}, the residue class \overline{2} is a zero divisor since \overline{2} \times \overline{2}=\overline{4}=\overline{0}.
  • The ring \mathbb{Z} of integers has no zero divisors except for 0.
  • A nilpotent element of a nonzero ring is always a two-sided zero divisor.
  • A idempotent element e\ne 1 of a ring is always a two-sided zero divisor, since e(1-e)=0=(1-e)e.
  • Examples of zero divisors in the ring of 2\times 2 matrices (over any nonzero ring) are shown here:
    \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} ,
    \begin{pmatrix}1&0\\0&0\end{pmatrix}\begin{pmatrix}0&0\\0&1\end{pmatrix}
=\begin{pmatrix}0&0\\0&1\end{pmatrix}\begin{pmatrix}1&0\\0&0\end{pmatrix}
=\begin{pmatrix}0&0\\0&0\end{pmatrix}.
  • A direct product of two or more nonzero rings always has nonzero zero divisors. For example, in R1 × R2 with each Ri nonzero, (1,0)(0,1) = (0,0), so (1,0) is a zero divisor.

One-sided zero-divisor[edit]

  • 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[edit]

  • The ring of integers modulo a prime number has no zero divisors except 0. Since every nonzero element is a unit, this ring is a field.

Properties[edit]

  • Left or right zero divisors can never be units, because if a is invertible and ax = 0, then 0 = a−10 = a−1ax = x, whereas x must be nonzero.

Zero as a zero divisor[edit]

There is no need for a separate convention regarding the case a = 0, because the definition applies also in this case:

  • If R is a ring other than the zero ring, then 0 is a (two-sided) zero divisor, because 0 · 1 = 0 and 1 · 0 = 0.
  • If R is the zero ring, in which 0 = 1, then 0 is not a zero divisor, because there is no nonzero element that when multiplied by 0 yields 0.

Such properties are needed in order to make the following general statements true:

  • In a commutative ring R, the set of non-zero-divisors is a multiplicative set in R. (This, in turn, is important for the definition of the total quotient ring.) The same is true of the set of non-left-zero-divisors and the set of non-right-zero-divisors in an arbitrary ring, commutative or not.
  • In a commutative ring R, the set of zero divisors is the union of the associated prime ideals of R.

Some references choose to exclude 0 as a zero divisor by convention, but then they must introduce exceptions in the two general statements just made.

Zero divisor on a module[edit]

Let R be a commutative ring, let M be an R-module, and let a be an element of R. One says that a is M-regular if the multiplication by a map M \stackrel{a}\to M is injective, and that a is a zero divisor on M otherwise.[3] The set of M-regular elements is a multiplicative set in R.[4]

Specializing the definitions of "M-regular" and "zero divisor on M" to the case M = R recovers the definitions of "regular" and "zero divisor" given earlier in this article.

See also[edit]

Notes[edit]

  1. ^ See Bourbaki, p. 98.
  2. ^ See Lanski (2005).
  3. ^ Matsumura, p. 12
  4. ^ Matsumura, p. 12

References[edit]