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| | Friends call her Herlinda but she doesn't like when people use her full moniker. His wife doesn't like it the way he does but what he really loves doing is play basketball and now he is intending to building an income with it. Debt collecting is generate income make a full time income. [http://search.Un.org/search?ie=utf8&site=un_org&output=xml_no_dtd&client=UN_Website_en&num=10&lr=lang_en&proxystylesheet=UN_Website_en&oe=utf8&q=Georgia&Submit=Go Georgia] is where her property is but her [http://search.Usa.gov/search?query=husband husband] wants them to help. See what's new on her website here: http://is.gd/bibijonesanal60890<br><br>My web page - [http://is.gd/bibijonesanal60890 bibi Jones] |
| 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. 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.
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| 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'''.
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| == Examples ==
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| <!-- it was valid, but nobody uses specifically Z×Z as a ring and hence, nobody cares about its zero divisors. In any way, generalized with the direct product example below. -->
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| * In the [[modular arithmetic|ring <math>\mathbb{Z}/4\mathbb{Z}</math>]], the residue class <math>\overline{2}</math> is a zero divisor since {{math|<math>\overline{2} \times \overline{2}=\overline{4}=\overline{0}</math>}}.
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| * The ring <math>\mathbb{Z}</math> of [[integer]]s has no zero divisors except for 0.
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| * A [[nilpotent]] element of a nonzero ring is always a two-sided zero divisor.
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| * 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>.
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| * 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\\
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| 2&2\end{pmatrix}</math>, because for instance <math>\begin{pmatrix}1&1\\
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| 2&2\end{pmatrix}\begin{pmatrix}1&1\\
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| -1&-1\end{pmatrix}=\begin{pmatrix}-2&1\\
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| -2&1\end{pmatrix}\begin{pmatrix}1&1\\
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| 2&2\end{pmatrix}=\begin{pmatrix}0&0\\
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| 0&0\end{pmatrix}.</math>
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| *Actually, the simplest example of a pair of zero divisor matrices is <math>
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| \begin{pmatrix}1&0\\0&0\end{pmatrix}
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| \begin{pmatrix}0&0\\0&1\end{pmatrix}
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| = | |
| \begin{pmatrix}0&0\\0&0\end{pmatrix}
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| = | |
| \begin{pmatrix}0&0\\0&1\end{pmatrix}
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| \begin{pmatrix}1&0\\0&0\end{pmatrix}</math>.
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| *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.
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| ===One-sided zero-divisor===
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| *Consider the ring of (formal) matrices <math>\begin{pmatrix}x&y\\0&z\end{pmatrix}</math> with <math>x,z\in\mathbb{Z}</math> and <math>y\in\mathbb{Z}/2\mathbb{Z}</math>. Then <math>\begin{pmatrix}x&y\\0&z\end{pmatrix}\begin{pmatrix}a&b\\0&c\end{pmatrix}=\begin{pmatrix}xa&xb+yc\\0&zc\end{pmatrix}</math> and <math>\begin{pmatrix}a&b\\0&c\end{pmatrix}\begin{pmatrix}x&y\\0&z\end{pmatrix}=\begin{pmatrix}xa&ya+zb\\0&zc\end{pmatrix}</math>. If <math>x\ne0\ne y</math>, then <math>\begin{pmatrix}x&y\\0&z\end{pmatrix}</math> is a left zero divisor [[iff]] <math>x</math> is even, since <math>\begin{pmatrix}x&y\\0&z\end{pmatrix}\begin{pmatrix}0&1\\0&0\end{pmatrix}=\begin{pmatrix}0&x\\0&0\end{pmatrix}</math>; and it is a right zero divisor iff <math>z</math> is even for similar reasons. If either of <math>x,z</math> is <math>0</math>, then it is a two-sided zero-divisor.
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| *Here is another example of a ring with an element that is a zero divisor on one side only. Let <math>S</math> be the set of all [[sequence (mathematics)|sequences]] of integers <math>(a1,a2,a3,...)</math>. Take for the ring all [[additive function|additive maps]] from <math>S</math> to <math>S</math>, with [[pointwise]] addition and [[function composition|composition]] as the ring operations. (That is, our ring is <math>\mathrm{End}(S)</math>, the '''[[endomorphism ring]]''' of the additive group <math>S</math>.) Three examples of elements of this ring are the '''right shift''' <math>R(a1,a2,a3,...)=(0,a1,a2,...)</math>, the '''left shift''' <math>L(a1,a2,a3,...)=(a2,a3,a4,...)</math>, and the '''projection map''' onto the first factor <math>P(a1,a2,a3,...)=(a1,0,0,...)</math>. All three of these [[additive function|additive maps]] are not zero, and the composites <math>LP</math> and <math>PR</math> are both zero, so <math>L</math> is a left zero divisor and <math>R</math> is a right zero divisor in the ring of additive maps from <math>S</math> to <math>S</math>. However, <math>L</math> is not a right zero divisor and <math>R</math> is not a left zero divisor: the composite <math>LR</math> is the identity. Note also that <math>RL</math> is a two-sided zero-divisor since <math>RLP=0=PRL</math>, while <math>LR=1</math> is not in any direction.
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| == Non-examples ==
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| * 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]].
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| * More generally, a [[division ring]] has no zero divisors except 0.
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| * A [[zero ring|nonzero]] [[commutative ring]] whose only zero divisor is 0 is called an [[integral domain]].
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| == Properties ==
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| * In the ring of {{mvar|n}}-by-{{mvar|n}} matrices over a [[field (mathematics)|field]], the left and right zero divisors coincide; they are precisely the [[singular matrix|singular matrices]]. In the ring of {{mvar|n}}-by-{{mvar|n}} matrices over an [[integral domain]], the zero divisors are precisely the matrices with [[determinant]] [[0 (number)|zero]].
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| * Left or right zero divisors can never be [[unit (ring theory)|unit]]s, because if {{mvar|a}} is invertible and {{math|1=''ax'' = 0}}, then {{math|1=0 = ''a''<sup>−1</sup>0 = ''a''<sup>−1</sup>''ax'' = ''x''}}, whereas {{math|''x''}} must be nonzero.
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| ==Zero as a zero divisor==
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| There is no need for a separate convention regarding the case {{math|1=''a'' = 0}}, because the definition applies also in this case:
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| * If {{mvar|R}} is a ring other than the [[zero ring]], then 0 is a (two-sided) zero divisor, because {{math|1=0 · 1 = 0}} and {{math|1=1 · 0 = 0}}.
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| * If {{mvar|R}} is the [[zero ring]], in which {{math|1=0 = 1}}, then 0 is not a zero divisor, because there is no ''nonzero'' element that when multiplied by 0 yields 0.
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| Such properties are needed in order to make the following general statements true:
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| * In a commutative ring {{mvar|R}}, the set of non-zero-divisors is a [[multiplicative set]] in {{mvar|R}}. (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.
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| * In a commutative ring {{mvar|R}}, the set of zero divisors is the union of the [[associated prime|associated prime ideals]] of {{mvar|R}}.
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| 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.
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| ==Zero divisor on a module==
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| Let {{mvar|R}} be a commutative ring, let {{mvar|M}} be an {{mvar|R}}-module, and let {{mvar|a}} be an element of {{mvar|R}}. One says that {{mvar|a}} is '''{{mvar|M}}-regular''' if the multiplication by {{mvar|a}} map <math>M \stackrel{a}\to M</math> is injective, and that {{mvar|a}} is a '''zero divisor on {{mvar|M}}''' otherwise.<ref>Matsumura, p. 12</ref> The set of {{mvar|M}}-regular elements is a [[multiplicative set]] in {{mvar|R}}.<ref>Matsumura, p. 12</ref>
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| Specializing the definitions of "{{mvar|M}}-regular" and "zero divisor on {{mvar|M}}" to the case {{mvar|M}} = {{mvar|R}} recovers the definitions of "regular" and "zero divisor" given earlier in this article.
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| ==See also==
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| * [[Zero-product property]]
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| * [[Glossary of commutative algebra]] (Exact zero divisor)
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| == Notes ==
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| <references/>
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| == References ==
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| * {{citation |author= [[N. Bourbaki]] |title=Algebra I, Chapters 1–3 |publisher=Springer-Verlag |year=1989}}.
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| * {{springer|title=Zero divisor|id=p/z099230}}
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| * {{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 }}
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| * {{citation |author= Charles Lanski |year=2005 |title=Concepts in Abstract Algebra |publisher=American Mathematical Soc. |page=342 }}
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| * {{citation |author=[[Hideyuki Matsumura]] |year=1980 |title=Commutative algebra, 2nd edition |publisher=The Benjamin/Cummings Publishing Company, Inc.}}
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| * {{MathWorld |title=Zero Divisor |urlname=ZeroDivisor }}
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| {{DEFAULTSORT:Zero Divisor}}
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| [[Category:Abstract algebra]]
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| [[Category:Ring theory]]
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| [[Category:Zero]]
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