# Difference between revisions of "Zero divisor"

en>Ebony Jackson (→Examples: simplified explanation) |
en>Cydebot m (Robot - Moving category Zero to Category:0 (number) per CFD at Wikipedia:Categories for discussion/Log/2014 November 19.) |
||

Line 1: | Line 1: | ||

{{refimprove|date=March 2010}} | {{refimprove|date=March 2010}} | ||

In [[abstract algebra]], an [[element (mathematics)|element]] {{mvar|a}} of a [[ring (algebra)|ring]] {{mvar|R}} is called a '''left zero divisor''' if there exists a nonzero {{mvar|x}} such that {{math|1=''ax'' = 0}}, or equivalently if the map from {{mvar|''R''}} to {{mvar|''R''}} | In [[abstract algebra]], an [[element (mathematics)|element]] {{mvar|a}} of a [[ring (algebra)|ring]] {{mvar|R}} is called a '''left zero divisor''' if there exists a nonzero {{mvar|x}} such that {{math|1=''ax'' = 0}}, or equivalently if the map from {{mvar|''R''}} to {{mvar|''R''}} that sends {{mvar|x}} to {{mvar|ax}} is not injective.<ref>See Bourbaki, p. 98.</ref> Similarly, an [[element (mathematics)|element]] {{mvar|a}} of a ring is called a '''right zero divisor''' if there exists a nonzero {{mvar|y}} such that {{math|1=''ya'' = 0}}. This is a partial case of [[divisibility (ring theory)|divisibility]] in rings. An element that is a left or a right zero divisor is simply called a '''zero divisor'''.<ref>See Lanski (2005).</ref> An element {{mvar|a}} that is both a left and a right zero divisor is called a '''two-sided zero divisor''' (the nonzero {{mvar|x}} such that {{math|1=''ax'' = 0}} may be different from the nonzero {{mvar|y}} such that {{math|1=''ya'' = 0}}). If the [[commutative ring|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'''. | 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'''. | ||

Line 10: | Line 10: | ||

* A [[nilpotent]] element of a nonzero ring is always a two-sided zero divisor. | * A [[nilpotent]] element of a nonzero ring is always a two-sided zero divisor. | ||

* A [[idempotent element]] <math>e\ne 1</math> of a ring is always a two-sided zero divisor, since <math>e(1-e)=0=(1-e)e</math>. | * A [[idempotent element]] <math>e\ne 1</math> of a ring is always a two-sided zero divisor, since <math>e(1-e)=0=(1-e)e</math>. | ||

* | * Examples of zero divisors in the [[matrix ring|ring of <math>2\times 2</math> matrices]] (over any [[zero ring|nonzero ring]]) are shown here: | ||

*:<math>\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} ,</math> | |||

2&2\end{pmatrix}\begin{pmatrix}1&1\\ | *:<math>\begin{pmatrix}1&0\\0&0\end{pmatrix}\begin{pmatrix}0&0\\0&1\end{pmatrix} | ||

-1&-1\end{pmatrix}=\begin{pmatrix}-2&1\\ | =\begin{pmatrix}0&0\\0&1\end{pmatrix}\begin{pmatrix}1&0\\0&0\end{pmatrix} | ||

-2&1\end{pmatrix}\begin{pmatrix}1&1\\ | =\begin{pmatrix}0&0\\0&0\end{pmatrix}</math>. | ||

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& | |||

\begin{pmatrix} | |||

\begin{pmatrix} | |||

*A [[product of rings|direct product]] of two or more [[zero ring|nonzero rings]] always has nonzero zero divisors. For example, in ''R''<sub>1</sub> × ''R''<sub>2</sub> with each ''R''<sub>''i''</sub> nonzero, (1,0)(0,1) = (0,0), so (1,0) is a zero divisor. | *A [[product of rings|direct product]] of two or more [[zero ring|nonzero rings]] always has nonzero zero divisors. For example, in ''R''<sub>1</sub> × ''R''<sub>2</sub> with each ''R''<sub>''i''</sub> nonzero, (1,0)(0,1) = (0,0), so (1,0) is a zero divisor. | ||

Line 33: | Line 23: | ||

== Non-examples == | == Non-examples == | ||

* The ring of integers [[modular arithmetic|modulo]] a [[prime number]] has no zero divisors except 0. | * The ring of integers [[modular arithmetic|modulo]] a [[prime number]] has no zero divisors except 0. Since every nonzero element is a [[unit (ring theory)|unit]], this ring is a [[field (mathematics)|field]]. | ||

* More generally, a [[division ring]] has no zero divisors except 0. | * More generally, a [[division ring]] has no zero divisors except 0. | ||

Line 72: | Line 62: | ||

* {{citation |author= [[N. Bourbaki]] |title=Algebra I, Chapters 1–3 |publisher=Springer-Verlag |year=1989}}. | * {{citation |author= [[N. Bourbaki]] |title=Algebra I, Chapters 1–3 |publisher=Springer-Verlag |year=1989}}. | ||

* {{springer|title=Zero divisor|id=p/z099230}} | * {{springer|title=Zero divisor|id=p/z099230}} | ||

* {{citation | * {{citation |year=2004 |title=Algebras, rings and modules |volume=Vol. 1 |publisher=Springer |isbn=1-4020-2690-0 |author-separator=, |author1 = Michiel Hazewinkel|author2 = Nadiya Gubareni|author3=Nadezhda Mikhaĭlovna Gubareni |author4=Vladimir V. Kirichenko. |authorlink1=Michiel Hazewinkel }} | ||

* {{citation |author= Charles Lanski |year=2005 |title=Concepts in Abstract Algebra |publisher=American Mathematical Soc. |page=342 }} | * {{citation |author= Charles Lanski |year=2005 |title=Concepts in Abstract Algebra |publisher=American Mathematical Soc. |page=342 }} | ||

* {{citation |author=[[Hideyuki Matsumura]] |year=1980 |title=Commutative algebra, 2nd edition |publisher=The Benjamin/Cummings Publishing Company, Inc.}} | * {{citation |author=[[Hideyuki Matsumura]] |year=1980 |title=Commutative algebra, 2nd edition |publisher=The Benjamin/Cummings Publishing Company, Inc.}} | ||

Line 80: | Line 70: | ||

[[Category:Abstract algebra]] | [[Category:Abstract algebra]] | ||

[[Category:Ring theory]] | [[Category:Ring theory]] | ||

[[Category: | [[Category:0 (number)]] |

## Latest revision as of 21:59, 28 November 2014

{{ 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

- In the ring , the residue class is a zero divisor since .
- The ring of integers has no zero divisors except for 0.
- A nilpotent element of a nonzero ring is always a two-sided zero divisor.
- A idempotent element of a ring is always a two-sided zero divisor, since .
- Examples of zero divisors in the ring of matrices (over any nonzero ring) are shown here:
- A direct product of two or more nonzero rings always has nonzero zero divisors. For example, in
*R*_{1}×*R*_{2}with each*R*_{i}nonzero, (1,0)(0,1) = (0,0), so (1,0) is a zero divisor.

### One-sided zero-divisor

- Consider the ring of (formal) matrices with and . Then and . If , then is a left zero divisor iff is even, since ; and it is a right zero divisor iff is even for similar reasons. If either of is , 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 be the set of all sequences of integers . Take for the ring all additive maps from to , with pointwise addition and composition as the ring operations. (That is, our ring is , the
**endomorphism ring**of the additive group .) Three examples of elements of this ring are the**right shift**, the**left shift**, and the**projection map**onto the first factor . All three of these additive maps are not zero, and the composites and are both zero, so is a left zero divisor and is a right zero divisor in the ring of additive maps from to . However, is not a right zero divisor and is not a left zero divisor: the composite is the identity. Note also that is a two-sided zero-divisor since , while is not in any direction.

## Non-examples

- The ring of integers modulo a prime number has no zero divisors except 0. Since every nonzero element is a unit, this ring is a field.

- More generally, a division ring has no zero divisors except 0.

- A nonzero commutative ring whose only zero divisor is 0 is called an integral domain.

## Properties

- In the ring of Template:Mvar-by-Template:Mvar matrices over a field, the left and right zero divisors coincide; they are precisely the singular matrices. In the ring of Template:Mvar-by-Template:Mvar 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 Template:Mvar is invertible and
*ax*= 0, then 0 =*a*^{−1}0 =*a*^{−1}*ax*=*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:

- In a commutative ring Template:Mvar, the set of non-zero-divisors is a multiplicative set in Template:Mvar. (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 ring Template:Mvar, the set of zero divisors is the union of the associated prime ideals of Template:Mvar.

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

- Zero-product property
- Glossary of commutative algebra (Exact zero divisor)

## Notes

## 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 }}