Editing Zero divisor (section)

== 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.