Nilpotent - Examples

Examples

This definition can be applied in particular to square matrices. The matrix

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

is nilpotent because A3 = 0. See nilpotent matrix for more.

In the factor ring Z/9Z, the equivalence class of 3 is nilpotent because 32 is congruent to 0 modulo 9.

Assume that two elements a, b in a (non-commutative) ring R satisfy ab = 0. Then the element c = ba is nilpotent (if non-zero) as c2 = (ba)2 = b(ab)a = 0. An example with matrices (for a, b):

$A = \begin{pmatrix} 0&1\\ 0&1 \end{pmatrix}, \;\; B =\begin{pmatrix} 0&1\\ 0&0 \end{pmatrix}.$

Here AB = 0, BA = B.

The ring of coquaternions contains a cone of nilpotents.

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