Zero divisor

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

• 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 only zero divisor of the ring \( \mathbb{Z} \)of integers is 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 R_i nonzero, (1,0)(0,1) = (0,0), so (1,0) is a zero divisor.

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 has no zero divisors other than 0. Since every nonzero element is a unit, this ring is a field.
• More generally, a division ring has no zero divisors except 0.
• A nonzero commutative ring whose only zero divisor is 0 is called an integral domain.

Properties

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

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 Noetherian 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 moduleSpecializing 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.

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.

Ring Theory 5: Zero Divisors and Integral Domains

See also

Zero-product property
Glossary of commutative algebra (Exact zero divisor)

Notes

See Bourbaki, p. 98.
See Lanski (2005).
Matsumura, p. 12

Matsumura, p. 12

References

N. Bourbaki (1989), Algebra I, Chapters 1–3, Springer-Verlag.
Hazewinkel, Michiel, ed. (2001), "Zero divisor", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Michiel Hazewinkel; Nadiya Gubareni; Nadezhda Mikhaĭlovna Gubareni; Vladimir V. Kirichenko. (2004), Algebras, rings and modules, Vol. 1, Springer, ISBN 1-4020-2690-0
Charles Lanski (2005), Concepts in Abstract Algebra, American Mathematical Soc., p. 342
Hideyuki Matsumura (1980), Commutative algebra, 2nd edition, The Benjamin/Cummings Publishing Company, Inc.
Weisstein, Eric W., "Zero Divisor", MathWorld.

Any non-trivial idempotent a is a zero divisor (because ab = 0 with neither a nor b being zero, where b = 1 − a). This shows that integral domains and division rings don't have such idempotents. Local rings also don't have such idempotents, but for a different reason. The only idempotent contained in the Jacobson radical of a ring is 0.

For associative algebras or Jordan algebras over a field, the Peirce decomposition is a decomposition of an algebra as a sum of eigenspaces of commuting idempotent elements.

Another method to obtain a field from a commutative ring R is taking the quotient R / m, where m is any maximal ideal of R. The above construction of F = E[X] / (p(X)), is an example, because the irreducibility of the polynomial p(X) is equivalent to the maximality of the ideal generated by this polynomial. Another example are the finite fields Fp = Z / pZ.