View source for Zero divisor

{{Short description|Ring element that can be multiplied by a non-zero element to equal 0}}
{{distinguish|Division by zero}}
{{Use American English|date = March 2019}}
In [[abstract algebra]], an [[element (mathematics)|element]] {{math|''a''}} of a [[ring (algebra)|ring]] {{math|''R''}} is called a '''left zero divisor''' if there exists a nonzero {{math|''x''}} in {{math|''R''}} such that {{math|1=''ax'' = 0}},<ref>{{citation |author= N. Bourbaki |author-link= N. Bourbaki |title=Algebra I, Chapters 1–3 |page=98 |publisher=Springer-Verlag |year=1989}}</ref> or equivalently if the [[function (mathematics)|map]] from {{math|''R''}} to {{math|''R''}} that sends {{math|''x''}} to {{math|''ax''}} is not [[injective]].{{efn|1=Since the map is not injective, we have {{math|1=''ax'' = ''ay''}}, in which {{math|''x''}} differs from {{math|''y''}}, and thus {{math|1=''a''(''x'' − ''y'') = 0}}.}}  Similarly, an element {{math|''a''}} of a ring is called a '''right zero divisor''' if there exists a nonzero {{math|''y''}} in {{math|''R''}} 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>{{citation |author= Charles Lanski |year=2005 |title=Concepts in Abstract Algebra |publisher=American Mathematical Soc. |page=342 }}</ref> An element&nbsp;{{math|''a''}} that is both a left and a right zero divisor is called a '''two-sided zero divisor''' (the nonzero {{math|''x''}} such that {{math|1=''ax'' = 0}} may be different from the nonzero {{math|''y''}} such that {{math|1=''ya'' = 0}}). If the ring is [[commutative ring|commutative]], then the left and right zero divisors are the same.
An element of a ring that is not a left zero divisor (respectively, not a right zero divisor) is called '''left regular''' or '''left cancellable''' (respectively, '''right regular''' or '''right cancellable''').
An element of a ring that is left and right cancellable, and is hence not a zero divisor, is called '''regular''' or '''cancellable''',{{refn|{{cite book|author=Nicolas Bourbaki|year=1998|title=Algebra I|publisher=[[Springer Science+Business Media]]|page=15}}}} or a '''non-zero-divisor'''.  A zero divisor that is nonzero is called a '''nonzero zero divisor''' or a '''nontrivial zero divisor'''.  A non-[[zero ring|zero]] ring with no nontrivial zero divisors is called a [[domain (ring theory)|domain]].
== 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 [[Ring (mathematics)|ring]] [[modular arithmetic|<math>\mathbb{Z}/4\mathbb{Z}</math>]], the residue class <math>\overline{2}</math> is a zero divisor since <math>\overline{2} \times \overline{2}=\overline{4}=\overline{0}</math>.
* The only zero divisor of the ring <math>\mathbb{Z}</math> of [[Integer#Algebraic properties|integers]] is <math>0</math>.
* A [[nilpotent]] element of a nonzero ring is always a two-sided zero divisor.
* An [[idempotent element (ring theory)|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>.
* The [[matrix ring|ring of ''n''&thinsp;×&thinsp;''n'' matrices]] over a [[field (mathematics)|field]] has nonzero zero divisors if ''n'' ≥ 2. Examples of zero divisors in the ring of 2&thinsp;×&thinsp;2 [[matrix (mathematics)|matrices]] (over any nonzero ring) are shown here:
<math display="block">\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> <math display="block">\begin{pmatrix}1&0\\0&0\end{pmatrix}\begin{pmatrix}0&0\\0&1\end{pmatrix}
=\begin{pmatrix}0&0\\0&1\end{pmatrix}\begin{pmatrix}1&0\\0&0\end{pmatrix}
=\begin{pmatrix}0&0\\0&0\end{pmatrix}.</math>
* A [[product of rings|direct product]] of two or more nonzero rings always has nonzero zero divisors.  For example, in <math>R_1 \times R_2</math> with each <math>R_i</math> nonzero, <math>(1,0)(0,1) = (0,0)</math>, so <math>(1,0)</math> is a zero divisor.
* Let <math>K</math> be a field and <math>G</math> be a [[group (mathematics)|group]].  Suppose that <math>G</math> has an element <math>g</math> of finite [[order (group theory)|order]] <math>n > 1</math>. Then in the [[group ring]] <math>K[G]</math> one has <math>(1-g)(1+g+ \cdots +g^{n-1})=1-g^{n}=0</math>, with neither factor being zero, so <math>1-g</math> is a nonzero zero divisor in <math>K[G]</math>.
=== 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 z</math>, then <math>\begin{pmatrix}x&y\\0&z\end{pmatrix}</math> is a left zero divisor [[if and only if]] <math>x</math> is [[parity (mathematics)|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 if and only if <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 (mathematics)|set]] of all [[sequence]]s of integers <math>(a_1,a_2,a_3,...)</math>.  Take for the ring all [[additive map]]s 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(a_1,a_2,a_3,...)=(0,a_1,a_2,...)</math>, the '''left shift''' <math>L(a_1,a_2,a_3,...)=(a_2,a_3,a_4,...)</math>, and the '''projection map''' onto the first factor <math>P(a_1,a_2,a_3,...)=(a_1,0,0,...)</math>.  All three of these 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. <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 nonzero zero divisors.  Since every nonzero element is a [[unit (ring theory)|unit]], this ring is a [[finite field]].
* More generally, a [[division ring]] has no nonzero zero divisors.
* A non-zero commutative ring whose only zero divisor is 0 is called an [[integral domain]].
== Properties ==
* In the ring of {{mvar|n}}&thinsp;×&thinsp;{{mvar|n}} matrices over a field, the left and right zero divisors coincide; they are precisely the [[singular matrix|singular matrices]]. In the ring of {{mvar|n}}&thinsp;×&thinsp;{{mvar|n}} matrices over an [[integral domain]], the zero divisors are precisely the matrices with [[determinant]] zero.
* Left or right zero divisors can never be [[unit (ring theory)|unit]]s, because if {{math|''a''}} is invertible and {{math|1=''ax'' = 0}} for some nonzero {{math|''x''}}, then {{math|1=0 = ''a''<sup>−1</sup>0 = ''a''<sup>−1</sup>''ax'' = ''x''}}, a contradiction.
* An element is [[Cancellation property|cancellable]] on the side on which it is regular.  That is, if {{math|''a''}} is a left regular, {{math|1=''ax'' = ''ay''}} implies that {{math|1=''x'' = ''y''}}, and similarly for right regular.
== Zero as a zero divisor ==
There is no need for a separate convention for the case {{math|1=''a'' = 0}}, because the definition applies also in this case:
* If {{math|''R''}} is a ring other than the [[zero ring]], then {{math|0}} is a (two-sided) zero divisor, because any nonzero element {{mvar|x}} satisfies {{math|1=0''x'' = 0 = ''x''&hairsp;0}}.
* If {{math|''R''}} is the zero ring, in which {{math|0 {{=}} 1}}, then {{math|0}} is not a zero divisor, because there is no ''nonzero'' element that when multiplied by {{math|0}} yields {{math|0}}.
Some references include or exclude {{math|0}} as a zero divisor in ''all'' rings by convention, but they then suffer from having to introduce exceptions in statements such as the following:
* In a commutative ring {{math|''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 [[noetherian ring]] {{math|''R''}}, the set of zero divisors is the [[union (set theory)|union]] of the [[associated prime|associated prime ideals]] of {{math|''R''}}.
== Zero divisor on a module ==
Let {{mvar|R}} be a commutative ring, let {{mvar|M}} be an {{mvar|R}}-[[Module (mathematics)|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 name=Matsumura-p12>{{citation |author=Hideyuki Matsumura |author-link=Hideyuki Matsumura |year=1980 |title=Commutative algebra, 2nd edition |publisher=The Benjamin/Cummings Publishing Company, Inc. |page=12}}</ref>  The set of {{mvar|M}}-regular elements is a [[multiplicative set]] in {{mvar|R}}.<ref name=Matsumura-p12/>
Specializing the definitions of "{{mvar|M}}-regular" and "zero divisor on {{mvar|M}}" to the case {{math|1=''M'' = ''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)
* [[Zero-divisor graph]]
* [[Sedenion]]s, which have zero divisors
== Notes ==
{{notelist}}
== References ==
{{reflist}}
== Further reading ==
* {{springer|title=Zero divisor|id=p/z099230}}
* {{citation |year=2004 |title=Algebras, rings and modules |volume=1 |publisher=Springer |isbn=1-4020-2690-0 |author1 = Michiel Hazewinkel|author2 = Nadiya Gubareni|author3=Nadezhda Mikhaĭlovna Gubareni |author4=Vladimir V. Kirichenko. |author-link1=Michiel Hazewinkel }}
* {{MathWorld |title=Zero Divisor |urlname=ZeroDivisor }}
[[Category:Abstract algebra]]
[[Category:Ring theory]]
[[Category:0 (number)]]
[[Category:Sedenions]]