en>Sammy1339: Noether was not at Bryn Mawr in 1929.

2014-11-02T22:51:00Z

Noether was not at Bryn Mawr in 1929.

en>Gilliam: Reverted edits by 108.199.41.103 (talk) to last version by 24.177.252.42

2014-02-01T18:22:25Z

Reverted edits by 108.199.41.103 (talk) to last version by 24.177.252.42

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	~~I like Bonsai~~. <br>I ~~to learn Bengali~~ in ~~my free time~~.<br><br>~~my web~~-~~site~~ [~~http~~://~~www~~.~~biz~~-~~suport~~.ro/~~resurse~~/~~articol/26~~-Te-ai-~~hotarat~~-sa-~~investesti~~-~~intr~~-o-~~afacere~~.-~~Care~~-~~sunt~~-~~pasii~~-~~urmatori FIFA coin Generator~~]		{{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. 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'''.

			== Examples ==
			<!-- 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. -->
			* 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>}}.
			* The ring <math>\mathbb{Z}</math> of [[integer]]s has no zero divisors except for 0.
			* 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>.
			* 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\\
			2&2\end{pmatrix}</math>, because for instance <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>
			*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.

			===One-sided zero-divisor===
			*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.
			*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.

			== 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]].

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

			* A [[zero ring\|nonzero]] [[commutative ring]] whose only zero divisor is 0 is called an [[integral domain]].

			== Properties ==

			* 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]].

			* 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.

			==Zero as a zero divisor==

			There is no need for a separate convention regarding the case {{math\|1=''a'' = 0}}, because the definition applies also in this case:
			* 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}}.
			* 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.

			Such properties are needed in order to make the following general statements true:
			* 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.
			* In a commutative ring {{mvar\|R}}, the set of zero divisors is the union of the [[associated prime\|associated prime ideals]] of {{mvar\|R}}.

			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 {{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>

			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.

			==See also==
			* [[Zero-product property]]
			* [[Glossary of commutative algebra]] (Exact zero divisor)

			== Notes ==
			<references/>

			== References ==
			* {{citation \|author= [[N. Bourbaki]] \|title=Algebra I, Chapters 1–3 \|publisher=Springer-Verlag \|year=1989}}.
			* {{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 \|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.}}
			* {{MathWorld \|title=Zero Divisor \|urlname=ZeroDivisor }}

			{{DEFAULTSORT:Zero Divisor}}
			[[Category:Abstract algebra]]
			[[Category:Ring theory]]
			[[Category:Zero]]

en>Headbomb: Various citation cleanup + AWB fixes using AWB (8062)

2012-08-16T02:16:01Z

Various citation cleanup + AWB fixes using AWB (8062)

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I like Bonsai. <br>I to learn Bengali in my free time.<br><br>my web-site [http://www.biz-suport.ro/resurse/articol/26-Te-ai-hotarat-sa-investesti-intr-o-afacere.-Care-sunt-pasii-urmatori FIFA coin Generator]

Hypercomplex number - Revision history

en>Sammy1339: Noether was not at Bryn Mawr in 1929.

en>Gilliam: Reverted edits by 108.199.41.103 (talk) to last version by 24.177.252.42

en>Headbomb: Various citation cleanup + AWB fixes using AWB (8062)