zero divisor

• In the ring $\backslash mathbb\{Z\}/4\backslash mathbb\{Z\}$, the residue class $\backslash overline\{2\}$ is a zero divisor since $\backslash overline\{2\}\; \backslash times\; \backslash overline\{2\}=\backslash overline\{4\}=\backslash overline\{0\}$.
• The only zero divisor of the ring $\backslash mathbb\{Z\}$ of integers is $0$.
• A nilpotent element of a nonzero ring is always a two-sided zero divisor.
• An idempotent element $e\backslash ne\; 1$ of a ring is always a two-sided zero divisor, since $e(1-e)=0=(1-e)e$.
• The ring of n × n matrices over a field has nonzero zero divisors if n ≥ 2. Examples of zero divisors in the ring of 2 × 2 matrices (over any nonzero ring) are shown here:

=\begin{pmatrix}0&0\\0&1\end{pmatrix}\begin{pmatrix}1&0\\0&0\end{pmatrix}

=\begin{pmatrix}0&0\\0&0\end{pmatrix}.

• A direct product of two or more nonzero rings always has nonzero zero divisors. For example, in $R\_1\; \backslash times\; R\_2$ with each $R\_i$ nonzero, $(1,0)(0,1)\; =\; (0,0)$, so $(1,0)$ is a zero divisor.
• Let $K$ be a field and $G$ be a group. Suppose that $G$ has an element $g$ of finite order $n\; >\; 1$. Then in the group ring $K[G]$ one has $(1-g)(1+g+\; \backslash cdots\; +g^\{n-1\})=1-g^\{n\}=0$, with neither factor being zero, so $1-g$ is a nonzero zero divisor in $K[G]$.

• Consider the ring of (formal) matrices with $x,z\backslash in\backslash mathbb\{Z\}$ and $y\backslash in\backslash mathbb\{Z\}/2\backslash mathbb\{Z\}$. Then and . If $x\backslash ne0\backslash ne\; z$, then is a left zero divisor if and only if $x$ is even, since , and it is a right zero divisor if and only if $z$ is even for similar reasons. If either of $x,z$ is $0$, 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 $S$ be the set of all sequences of integers $(a\_1,a\_2,a\_3,...)$. Take for the ring all additive maps from $S$ to $S$, with pointwise addition and composition as the ring operations. (That is, our ring is $\backslash mathrm\{End\}(S)$, the endomorphism ring of the additive group $S$.) Three examples of elements of this ring are the right shift $R(a\_1,a\_2,a\_3,...)=(0,a\_1,a\_2,...)$, the left shift $L(a\_1,a\_2,a\_3,...)=(a\_2,a\_3,a\_4,...)$, and the projection 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 $LP$ and $PR$ are both zero, so $L$ is a left zero divisor and $R$ is a right zero divisor in the ring of additive maps from $S$ to $S$. However, $L$ is not a right zero divisor and $R$ is not a left zero divisor: the composite $LR$ is the identity. $RL$ is a two-sided zero-divisor since $RLP=0=PRL$, while $LR=1$ is not in any direction.

• The ring of integers modulo a prime number has no nonzero zero divisors. Since every nonzero element is a 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.

• In the ring of {{mvar|n}} × {{mvar|n}} matrices over a field, the left and right zero divisors coincide; they are precisely the singular matrices. In the ring of {{mvar|n}} × {{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 units, because if {{math|a}} is invertible and {{math|1=ax = 0}} for some nonzero {{math|x}}, then {{math|1=0 = a⁻¹0 = a⁻¹ax = x}}, a contradiction.
• An element is 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.

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=0x = 0 = x 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 of the associated prime ideals of {{math|R}}.

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 $M\; \backslash ,\backslash stackrel\{a\}\backslash to\backslash ,\; M$ is injective, and that {{mvar|a}} is a zero divisor on {{mvar|M}} otherwise.{{citation |author=Hideyuki Matsumura |author-link=Hideyuki Matsumura |year=1980 |title=Commutative algebra, 2nd edition |publisher=The Benjamin/Cummings Publishing Company, Inc. |page=12}} The set of {{mvar|M}}-regular elements is a multiplicative set in {{mvar|R}}.

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.

• Zero-product property
• Glossary of commutative algebra (Exact zero divisor)
• Zero-divisor graph
• Sedenions, which have zero divisors

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{{reflist}}

• {{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