Difference between revisions of "Zero divisor"

From formulasearchengine
Jump to navigation Jump to search
en>Ebony Jackson
(→‎Examples: simplified explanation)
en>Cydebot
 
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''}} sending {{mvar|x}} to {{mvar|ax}} is not injective.<ref>See Bourbaki, p.&nbsp;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&nbsp;{{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.  
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.&nbsp;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&nbsp;{{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>.
* An example of a zero divisor in the [[matrix ring|ring of <math>2\times 2</math> matrices]] (over any [[zero ring|nonzero ring]]) is the [[matrix (mathematics)|matrix]] <math>\begin{pmatrix}1&1\\
* 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:
2&2\end{pmatrix}</math>, because for instance <math>\begin{pmatrix}1&1\\
*:<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}.</math>
*Actually, the simplest example of a pair of zero divisor matrices is <math>
\begin{pmatrix}1&0\\0&0\end{pmatrix}
\begin{pmatrix}0&0\\0&1\end{pmatrix}
=
\begin{pmatrix}0&0\\0&0\end{pmatrix}
=
\begin{pmatrix}0&0\\0&1\end{pmatrix}
\begin{pmatrix}1&0\\0&0\end{pmatrix}</math>.
*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.  In fact, this ring is a [[field (mathematics)|field]], since every nonzero element is a [[unit (ring theory)|unit]].
* 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 |author=[[Michiel Hazewinkel]], Nadiya Gubareni, Nadezhda Mikhaĭlovna Gubareni, Vladimir V. Kirichenko. |year=2004 |title=Algebras, rings and modules |volume=Vol. 1 |publisher=Springer |isbn=1-4020-2690-0 }}
* {{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:Zero]]
[[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

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 }}