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.

Examples

One-sided zero-divisor

Non-examples

Properties

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

Notes

  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

References

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