Zero divisor

From formulasearchengine
Jump to navigation Jump to search

{{ safesubst:#invoke:Unsubst||$N=Refimprove |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} In abstract algebra, an element Template:Mvar of a ring Template:Mvar is called a left zero divisor if there exists a nonzero Template:Mvar such that ax = 0, or equivalently if the map from Template:Mvar to Template:Mvar that sends Template:Mvar to Template:Mvar is not injective.[1] Similarly, an element Template:Mvar of a ring is called a right zero divisor if there exists a nonzero Template:Mvar 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 Template:Mvar that is both a left and a right zero divisor is called a two-sided zero divisor (the nonzero Template:Mvar such that ax = 0 may be different from the nonzero Template:Mvar 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

One-sided zero-divisor

Non-examples

Properties

  • Left or right zero divisors can never be units, because if Template:Mvar 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 Template:Mvar 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 Template:Mvar 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:

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 Template:Mvar be a commutative ring, let Template:Mvar be an Template:Mvar-module, and let Template:Mvar be an element of Template:Mvar. One says that Template:Mvar is Template:Mvar-regular if the multiplication by Template:Mvar map is injective, and that Template:Mvar is a zero divisor on Template:Mvar otherwise.[3] The set of Template:Mvar-regular elements is a multiplicative set in Template:Mvar.[4]

Specializing the definitions of "Template:Mvar-regular" and "zero divisor on Template:Mvar" to the case Template:Mvar = Template:Mvar recovers the definitions of "regular" and "zero divisor" given earlier in this article.

See also

Notes

  1. See Bourbaki, p. 98.
  2. See Lanski (2005).
  3. Matsumura, p. 12
  4. Matsumura, p. 12

References

  • {{#invoke:citation/CS1|citation

|CitationClass=citation }}.

  • {{#invoke:citation/CS1|citation

|CitationClass=citation }}

  • {{#invoke:citation/CS1|citation

|CitationClass=citation }}

  • {{#invoke:citation/CS1|citation

|CitationClass=citation }}

  • {{#invoke:citation/CS1|citation

|CitationClass=citation }}