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,[1] or equivalently if the map from R to R that sends x to ax is not injective.[2] 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.[3] 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. If there are no nontrivial zero divisors in R, then R is a domain.


One-sided zero-divisor



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:

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

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

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 is injective, and that a is a zero divisor on M otherwise.[4] The set of M-regular elements is a multiplicative set in R.[5]

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


  1. See Bourbaki, p. 98.
  2. Since the map is not injective, we have ax = ay, in which x differs from y, and thus a(x-y) = 0.
  3. See Lanski (2005).
  4. Matsumura, p. 12
  5. Matsumura, p. 12


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