In abstract algebra, an element a of a ring R is called a left zero divisor if there exists a nonzero x in R such that ax = 0,^[1] or equivalently if the map from R to R that sends x to ax is not injective.^[a] Similarly, an element a of a ring is called a right zero divisor if there exists a nonzero y in R 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 left zero divisor (respectively, not a right zero divisor) is called left regular or left cancellable (respectively, right regular or right cancellable). An element of a ring that is left and right cancellable, and is hence not a zero divisor, is called regular or cancellable,^[3] or a non-zero-divisor. (N.B.: In "non-zero-divisor", the prefix "non-" is understood to modify "zero-divisor" as a whole rather than just the word "zero". In some texts, "zero divisor" is written as "zerodivisor" and "non-zero-divisor" as "nonzerodivisor"^[4] or "non-zerodivisor"^[5] for clarity.) A zero divisor that is nonzero is called a nonzero zero divisor or a nontrivial zero divisor. A non-zero ring with no nontrivial zero divisors is called a domain.

• In the ring 👁 {\displaystyle \mathbb {Z} /4\mathbb {Z} }
  , the residue class 👁 {\displaystyle {\overline {2}}}
  is a zero divisor since 👁 {\displaystyle {\overline {2}}\times {\overline {2}}={\overline {4}}={\overline {0}}}
  .
• The only zero divisor of the ring 👁 {\displaystyle \mathbb {Z} }
  of integers is 👁 {\displaystyle 0}
  .
• A nilpotent element of a nonzero ring is always a two-sided zero divisor.
• An idempotent element 👁 {\displaystyle e\neq 1}
  of a ring is always a two-sided zero divisor, since 👁 {\displaystyle 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:

👁 {\displaystyle {\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}},}
👁 {\displaystyle {\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 👁 {\displaystyle R_{1}\times R_{2}}
  with each 👁 {\displaystyle R_{i}}
  nonzero, 👁 {\displaystyle (1,0)(0,1)=(0,0)}
  , so 👁 {\displaystyle (1,0)}
  is a zero divisor.
• Let 👁 {\displaystyle K}
  be a field and 👁 {\displaystyle G}
  be a group. Suppose that 👁 {\displaystyle G}
  has an element 👁 {\displaystyle g}
  of finite order 👁 {\displaystyle n>1}
  . Then in the group ring 👁 {\displaystyle K[G]}
  one has 👁 {\displaystyle (1-g)(1+g+\cdots +g^{n-1})=1-g^{n}=0}
  , with neither factor being zero, so 👁 {\displaystyle 1-g}
  is a nonzero zero divisor in 👁 {\displaystyle K[G]}
  .

• Consider the ring of (formal) matrices 👁 {\displaystyle {\begin{pmatrix}x&y\\0&z\end{pmatrix}}}
  with 👁 {\displaystyle x,z\in \mathbb {Z} }
  and 👁 {\displaystyle y\in \mathbb {Z} /2\mathbb {Z} }
  . Then 👁 {\displaystyle {\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 👁 {\displaystyle {\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 👁 {\displaystyle x\neq 0\neq z}
  , then 👁 {\displaystyle {\begin{pmatrix}x&y\\0&z\end{pmatrix}}}
  is a left zero divisor if and only if 👁 {\displaystyle x}
  is even, since 👁 {\displaystyle {\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 👁 {\displaystyle z}
  is even for similar reasons. If either of 👁 {\displaystyle x,z}
  is 👁 {\displaystyle 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 👁 {\displaystyle S}
  be the set of all sequences of integers 👁 {\displaystyle (a_{1},a_{2},a_{3},...)}
  . Take for the ring all additive maps from 👁 {\displaystyle S}
  to 👁 {\displaystyle S}
  , with pointwise addition and composition as the ring operations. (That is, our ring is 👁 {\displaystyle \mathrm {End} (S)}
  , the endomorphism ring of the additive group 👁 {\displaystyle S}
  .) Three examples of elements of this ring are the right shift 👁 {\displaystyle R(a_{1},a_{2},a_{3},...)=(0,a_{1},a_{2},...)}
  , the left shift 👁 {\displaystyle L(a_{1},a_{2},a_{3},...)=(a_{2},a_{3},a_{4},...)}
  , and the projection map onto the first factor 👁 {\displaystyle P(a_{1},a_{2},a_{3},...)=(a_{1},0,0,...)}
  . All three of these additive maps are not zero, and the composites 👁 {\displaystyle LP}
  and 👁 {\displaystyle PR}
  are both zero, so 👁 {\displaystyle L}
  is a left zero divisor and 👁 {\displaystyle R}
  is a right zero divisor in the ring of additive maps from 👁 {\displaystyle S}
  to 👁 {\displaystyle S}
  . However, 👁 {\displaystyle L}
  is not a right zero divisor and 👁 {\displaystyle R}
  is not a left zero divisor: the composite 👁 {\displaystyle LR}
  is the identity. 👁 {\displaystyle RL}
  is a two-sided zero-divisor since 👁 {\displaystyle RLP=0=PRL}
  , while 👁 {\displaystyle LR=1}
  is not in any direction.

• 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.

• In the ring of n × n matrices over a field, the left and right zero divisors coincide; they are precisely the singular matrices. In the ring of n × 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 for some nonzero x, then 0 = a⁻¹0 = a⁻¹ax = x, a contradiction.
• An element is cancellable on the side on which it is regular. That is, if a is a left regular, ax = ay implies that x = y, and similarly for right regular.

There is no need for a separate convention for 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 any nonzero element x satisfies 0x = 0 = x 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.

Some references include or exclude 0 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 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.

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 👁 {\displaystyle M\,{\stackrel {a}{\to }}\,M}
is injective, and that a is a zero divisor on M otherwise.^[6] The set of M-regular elements is a multiplicative set in R.^[6]

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.

• Zero-product property
• Glossary of commutative algebra (Exact zero divisor)
• Zero-divisor graph
• Sedenions, which have zero divisors

• "Zero divisor", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Michiel Hazewinkel; Nadiya Gubareni; Nadezhda Mikhaĭlovna Gubareni; Vladimir V. Kirichenko. (2004), Algebras, rings and modules, vol. 1, Springer, ISBN 1-4020-2690-0