Nilpotent

In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0.

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

:A = egin{pmatrix}

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0&1&0\

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0&0&1\

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0&0&0end{pmatrix}

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

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• In the factor ring Z/9Z, the 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_1 = egin{pmatrix}

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0&1\

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0&1

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end{pmatrix}, ;;

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A_2 =egin{pmatrix}

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0&1\

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0&0

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end{pmatrix} .

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: Here A_1A_2=0,; A_2A_1=A_2 .

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• The ring of coquaternions contains a cone of nilpotents.

Ring: A ring is usually anything resembling a circle, or a noise that cycles rapidly. See:...

Integer: The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. The set of all integers is usually denoted in mathematics by Z (or Z in blackboard bold, mathbb{Z}), which stands for Zahlen (German for "numbers"). The...