Examples
- This definition can be applied in particular to square matrices. The matrix
-
- 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):
-
- Here AB = 0, BA = B.
- The ring of coquaternions contains a cone of nilpotents.
Read more about this topic: Nilpotent
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