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Zero divisor

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Ring element that can be multiplied by a nonzero element to equal 0

Not to be confused withDivision by zero.

Inabstract algebra, anelementa of aringR is called aleft zero divisor if there exists a nonzerox inR such thatax = 0,^[1] or equivalently if themap fromR toR that sendsx toax is notinjective.^[a] Similarly, an elementa of a ring is called aright zero divisor if there exists a nonzeroy inR such thatya = 0. This is a partial case ofdivisibility in rings. An element that is a left or a right zero divisor is simply called azero divisor.^[2] An element a that is both a left and a right zero divisor is called atwo-sided zero divisor (the nonzerox such thatax = 0 may be different from the nonzeroy such thatya = 0). If the ring iscommutative, 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 calledleft regular orleft cancellable (respectively,right regular orright cancellable).An element of a ring that is left and right cancellable, and is hence not a zero divisor, is calledregular orcancellable,^[3] or anon-zero-divisor. (N.B.: In "non-zero-divisor", the prefix "non-" is understood to modify "zero-divisor" as a whole rather than just the word "zero". In some texts, "zero divisor" is written as "zerodivisor" and "non-zero-divisor" as "nonzerodivisor"^[4] or "non-zerodivisor"^[5] for clarity.) A zero divisor that is nonzero is called anonzero zero divisor or anontrivial zero divisor. A non-zero ring with no nontrivial zero divisors is called adomain.

Examples

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In thering $\mathbb {Z} /4\mathbb {Z}$ , the residue class ${\overline {2}}$ is a zero divisor since ${\overline {2}}\times {\overline {2}}={\overline {4}}={\overline {0}}$ .
The only zero divisor of the ring $\mathbb {Z}$ ofintegers is $0 {\displaystyle 0}$ .
Anilpotent element of a nonzero ring is always a two-sided zero divisor.
Anidempotent element $e\neq 1$ of a ring is always a two-sided zero divisor, since $e(1-e)=0=(1-e)e$ .
Thering ofn ×n matrices over afield has nonzero zero divisors ifn ≥ 2. Examples of zero divisors in the ring of 2 × 2matrices (over any nonzero ring) are shown here:

${\begin{pmatrix}1&1\\2&2\end{pmatrix}}{\begin{pmatrix}1&1\\-1&-1\end{pmatrix}}={\begin{pmatrix}-2&1\\-2&1\end{pmatrix}}{\begin{pmatrix}1&1\\2&2\end{pmatrix}}={\begin{pmatrix}0&0\\0&0\end{pmatrix}},$ ${\begin{pmatrix}1&0\\0&0\end{pmatrix}}{\begin{pmatrix}0&0\\0&1\end{pmatrix}}={\begin{pmatrix}0&0\\0&1\end{pmatrix}}{\begin{pmatrix}1&0\\0&0\end{pmatrix}}={\begin{pmatrix}0&0\\0&0\end{pmatrix}}.$

Adirect product of two or more nonzero rings always has nonzero zero divisors. For example, in $R_{1}\times R_{2}$ with each $R_{i}$ nonzero, $(1,0)(0,1)=(0,0)$ , so $(1,0)$ is a zero divisor.
Let $K {\displaystyle K}$ be a field and $G {\displaystyle G}$ be agroup. Suppose that $G {\displaystyle G}$ has an element $g {\displaystyle g}$ of finiteorder $n>1$ . Then in thegroup ring $K[G]$ one has $(1-g)(1+g+\cdots +g^{n-1})=1-g^{n}=0$ , with neither factor being zero, so $1-g$ is a nonzero zero divisor in $K[G]$ .

One-sided zero-divisor

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Consider the ring of (formal) matrices ${\begin{pmatrix}x&y\\0&z\end{pmatrix}}$ with $x,z\in \mathbb {Z}$ and $y\in \mathbb {Z} /2\mathbb {Z}$ . Then ${\begin{pmatrix}x&y\\0&z\end{pmatrix}}{\begin{pmatrix}a&b\\0&c\end{pmatrix}}={\begin{pmatrix}xa&xb+yc\\0&zc\end{pmatrix}}$ and ${\begin{pmatrix}a&b\\0&c\end{pmatrix}}{\begin{pmatrix}x&y\\0&z\end{pmatrix}}={\begin{pmatrix}xa&ya+zb\\0&zc\end{pmatrix}}$ . If $x\neq 0\neq z$ , then ${\begin{pmatrix}x&y\\0&z\end{pmatrix}}$ is a left zero divisorif and only if $x {\displaystyle x}$ iseven, since ${\begin{pmatrix}x&y\\0&z\end{pmatrix}}{\begin{pmatrix}0&1\\0&0\end{pmatrix}}={\begin{pmatrix}0&x\\0&0\end{pmatrix}}$ , and it is a right zero divisor if and only if $z {\displaystyle z}$ is even for similar reasons. If either of $x, z {\displaystyle x,z}$ is $0 {\displaystyle 0}$ , 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 $S {\displaystyle S}$ be theset of allsequences of integers $(a_{1},a_{2},a_{3},...)$ . Take for the ring alladditive maps from $S {\displaystyle S}$ to $S {\displaystyle S}$ , withpointwise addition andcomposition as the ring operations. (That is, our ring is $\mathrm {End} (S)$ , theendomorphism ring of the additive group $S {\displaystyle S}$ .) Three examples of elements of this ring are theright shift $R(a_{1},a_{2},a_{3},...)=(0,a_{1},a_{2},...)$ , theleft shift $L(a_{1},a_{2},a_{3},...)=(a_{2},a_{3},a_{4},...)$ , and theprojection map onto the first factor $P(a_{1},a_{2},a_{3},...)=(a_{1},0,0,...)$ . All three of these additive maps are not zero, and the composites $L P {\displaystyle LP}$ and $P R {\displaystyle PR}$ are both zero, so $L {\displaystyle L}$ is a left zero divisor and $R {\displaystyle R}$ is a right zero divisor in the ring of additive maps from $S {\displaystyle S}$ to $S {\displaystyle S}$ . However, $L {\displaystyle L}$ is not a right zero divisor and $R {\displaystyle R}$ is not a left zero divisor: the composite $L R {\displaystyle LR}$ is the identity. $R L {\displaystyle RL}$ is a two-sided zero-divisor since $RLP=0=PRL$ , while $LR=1$ is not in any direction.

Non-examples

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The ring of integersmodulo aprime number has no nonzero zero divisors. Since every nonzero element is aunit, this ring is afinite field.
More generally, adivision ring has no nonzero zero divisors.
A nonzero commutative ring whose only zero divisor is 0 is called anintegral domain.

Properties

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In the ring ofn ×n matrices over a field, the left and right zero divisors coincide; they are precisely thesingular matrices. In the ring ofn ×n matrices over anintegral domain, the zero divisors are precisely the matrices withdeterminant zero.
Left or right zero divisors can never beunits, because ifa is invertible andax = 0 for some nonzerox, then0 =a⁻¹0 =a⁻¹ax =x, a contradiction.
An element iscancellable on the side on which it is regular. That is, ifa is a left regular,ax =ay implies thatx =y, and similarly for right regular.

Zero as a zero divisor

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There is no need for a separate convention for the casea = 0, because the definition applies also in this case:

IfR is a ring other than thezero ring, then0 is a (two-sided) zero divisor, because any nonzero elementx satisfies0x = 0 =x 0.
IfR is the zero ring, in which0 = 1, then0 is not a zero divisor, because there is nononzero element that when multiplied by0 yields0.

Some references include or exclude0 as a zero divisor inall rings by convention, but they then suffer from having to introduce exceptions in statements such as the following:

In a commutative ringR, the set of non-zero-divisors is amultiplicative set inR. (This, in turn, is important for the definition of thetotal 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 commutativenoetherian ringR, the set of zero divisors is theunion of theassociated prime ideals ofR.

Zero divisor on a module

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LetR be a commutative ring, letM be anR-module, and leta be an element ofR. One says thata isM-regular if the "multiplication bya" map $M\,{\stackrel {a}{\to }}\,M$ is injective, and thata is azero divisor onM otherwise.^[6] The set ofM-regular elements is amultiplicative set inR.^[6]

Specializing the definitions of "M-regular" and "zero divisor onM" to the caseM =R recovers the definitions of "regular" and "zero divisor" given earlier in this article.

Notes

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^Since the map is not injective, we haveax =ay, in whichx differs fromy, and thusa(x −y) = 0.

References

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^N. Bourbaki (1989),Algebra I, Chapters 1–3, Springer-Verlag, p. 98
^Charles Lanski (2005),Concepts in Abstract Algebra, American Mathematical Soc., p. 342
^Nicolas Bourbaki (1998).Algebra I.Springer Science+Business Media. p. 15.
^"Non zero-divisors | Stacks Project Blog". 2012-05-10. Retrieved2025-07-20.
^Reid, Miles (1995).Undergraduate commutative algebra. London Mathematical Society student texts. Cambridge ; New York: Cambridge University Press.ISBN 978-0-521-45255-7.
^^a ^bHideyuki Matsumura (1980),Commutative algebra, 2nd edition, The Benjamin/Cummings Publishing Company, Inc., p. 12

Movatterモバイル変換

Zero divisor

Examples

One-sided zero-divisor

Non-examples

Properties

Zero as a zero divisor

Zero divisor on a module

See also

Notes

References

Further reading