Zero divisor

Template:Short description Template:Distinguish Template:Use American English In abstract algebra, an element Template:Math of a ring Template:Math is called a left zero divisor if there exists a nonzero Template:Math in Template:Math such that Template:Math,<ref>Template:Citation</ref> or equivalently if the map from Template:Math to Template:Math that sends Template:Math to Template:Math is not injective.Template:Efn Similarly, an element Template:Math of a ring is called a right zero divisor if there exists a nonzero Template:Math in Template:Math such that Template:Math. This is a partial case of divisibility in rings. An element that is a left or a right zero divisor is simply called a zero divisor.<ref>Template:Citation</ref> An element Template:Math that is both a left and a right zero divisor is called a two-sided zero divisor (the nonzero Template:Math such that Template:Math may be different from the nonzero Template:Math such that Template:Math). If the ring is 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,Template:Refn 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 with no nontrivial zero divisors is called a domain.

ExamplesEdit

In the ring <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 integers is <math>0</math>.
A nilpotent element of a nonzero ring is always a two-sided zero divisor.
An 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 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:

<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 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. Suppose that <math>G</math> has an element <math>g</math> of finite 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-divisorEdit

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 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 of all sequences of integers <math>(a_1,a_2,a_3,...)</math>. Take for the ring all additive maps from <math>S</math> to <math>S</math>, with pointwise addition and 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-examplesEdit

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.

PropertiesEdit

In the ring of Template:Mvar × Template:Mvar matrices over a field, the left and right zero divisors coincide; they are precisely the singular matrices. In the ring of Template:Mvar × Template:Mvar 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 Template:Math is invertible and Template:Math for some nonzero Template:Math, then Template:Math, a contradiction.
An element is cancellable on the side on which it is regular. That is, if Template:Math is a left regular, Template:Math implies that Template:Math, and similarly for right regular.

Zero as a zero divisorEdit

There is no need for a separate convention for the case Template:Math, because the definition applies also in this case:

If Template:Math is a ring other than the zero ring, then Template:Math is a (two-sided) zero divisor, because any nonzero element Template:Mvar satisfies Template:Math.
If Template:Math is the zero ring, in which Template:Math, then Template:Math is not a zero divisor, because there is no nonzero element that when multiplied by Template:Math yields Template:Math.

Some references include or exclude Template:Math 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 Template:Math, the set of non-zero-divisors is a multiplicative set in Template:Mvar. (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 Template:Math, the set of zero divisors is the union of the associated prime ideals of Template:Math.

Zero divisor on a moduleEdit

Let Template:Mvar be a commutative ring, let Template:Mvar be an Template:Mvar-module, and let Template:Mvar be an element of Template:Mvar. One says that Template:Mvar is Template:Mvar-regular if the "multiplication by Template:Mvar" map <math>M \,\stackrel{a}\to\, M</math> is injective, and that Template:Mvar is a zero divisor on Template:Mvar otherwise.<ref name=Matsumura-p12>Template:Citation</ref> The set of Template:Mvar-regular elements is a multiplicative set in Template:Mvar.<ref name=Matsumura-p12/>

Specializing the definitions of "Template:Mvar-regular" and "zero divisor on Template:Mvar" to the case Template:Math recovers the definitions of "regular" and "zero divisor" given earlier in this article.

NotesEdit

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ReferencesEdit

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