Editing Zero divisor (section)

=== 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.