Zero divisor

• 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. • An idempotent element e\ne 1 of a ring is always a two-sided zero divisor, since e(1-e)=0=(1-e)e. • The ring of n × n matrices over a field has nonzero zero divisors if n ≥ 2. Examples of zero divisors in the ring of 2 × 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 R_1 \times R_2 with each R_i nonzero, (1,0)(0,1) = (0,0), so (1,0) is a zero divisor. • Let K be a field and G be a group. Suppose that G has an element g of finite order n > 1. Then in the group ring K[G] one has (1-g)(1+g+ \cdots +g^{n-1})=1-g^{n}=0, with neither factor being zero, so 1-g is a nonzero zero divisor in K[G]. 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 z, then \begin{pmatrix}x&y\\0&z\end{pmatrix} is a left zero divisor if and only if 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 if and only if 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 (a_1,a_2,a_3,...). 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(a_1,a_2,a_3,...)=(0,a_1,a_2,...), the left shift L(a_1,a_2,a_3,...)=(a_2,a_3,a_4,...), and the projection map onto the first factor P(a_1,a_2,a_3,...)=(a_1,0,0,...). 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. 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 nonzero zero divisors. Since every nonzero element is a unit, this ring is a finite field. • More generally, a division ring has no nonzero zero divisors. • A nonzero commutative ring whose only zero divisor is 0 is called an integral domain. == Properties ==

• In the ring of × matrices over a field, the left and right zero divisors coincide; they are precisely the singular matrices. In the ring of × 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 is invertible and for some nonzero , then , a contradiction. • An element is cancellable on the side on which it is regular. That is, if is a left regular, implies that , and similarly for right regular. == Zero as a zero divisor ==

There is no need for a separate convention for the case , because the definition applies also in this case: • If is a ring other than the zero ring, then is a (two-sided) zero divisor, because any nonzero element satisfies . • If is the zero ring, in which , then is not a zero divisor, because there is no nonzero element that when multiplied by yields . Some references include or exclude as a zero divisor in all rings by convention, but they then suffer from having to introduce exceptions in statements such as the following: • In a commutative ring , the set of non-zero-divisors is a multiplicative set in . (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 , the set of zero divisors is the union of the associated prime ideals of . == Zero divisor on a module ==

Let be a commutative ring, let be an -module, and let be an element of . One says that is -regular if the "multiplication by " map M \,\stackrel{a}\to\, M is injective, and that is a zero divisor on otherwise. The set of -regular elements is a multiplicative set in . Specializing the definitions of "-regular" and "zero divisor on " to the case recovers the definitions of "regular" and "zero divisor" given earlier in this article. == See also ==