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Nilpotent

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

Contents

1 Examples

2 Properties

3 Nilpotency in physics

4 References

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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.

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):

$A_1 = \begin{pmatrix} 0&1\\ 0&1 \end{pmatrix}, \;\; A_2 =\begin{pmatrix} 0&1\\ 0&0 \end{pmatrix} \ .$

Here $A_1A_2=0,\; A_2A_1=A_2$ .

The ring of coquaternions contains a cone of nilpotents.

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Properties

No nilpotent element can be a unit (except in the trivial ring {0} which has only a single element 0 = 1). All non-zero nilpotent elements are zero divisors.

An n-by-n matrix A with entries from a field is nilpotent if and only if its characteristic polynomial is Tⁿ, which is the case if and only if Aⁿ = 0.

The nilpotent elements from a commutative ring form an ideal; this is a consequence of the binomial theorem. This ideal is the nilradical of the ring. Every nilpotent element in a commutative ring is contained in every prime ideal of that ring, and in fact the intersection of all these prime ideals is equal to the nilradical.

If x is nilpotent, then 1 − x is a unit, because xⁿ = 0 entails

(1 − x) (1 + x + x² + ... + xⁿ⁻¹) = 1 − xⁿ = 1.

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Nilpotency in physics

An operator $Q$ that satisfies $Q 2 = 0$ is nilpotent. The BRST charge is an important example in physics.

As linear operators form an associative algebra and thus a ring, this is a special case of the initial definition. More generally, in view of the above definitions, an operator Q is nilpotent if there is n∈N such that Qⁿ=o (the zero function). Thus, a linear map is nilpotent iff it has a nilpotent matrix in some basis. Another example for this is the exterior derivative (again with n=2). Both are linked, also through supersymmetry and Morse theory, as shown by Edward Witten in a celebrated article.

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References

E Witten, Supersymmetry and Morse theory. J.Diff.Geom.17:661-692,1982.
A. Rogers, The topological particle and Morse theory, Class. Quantum Grav. 17:3703-3714,2000 DOI:10.1088/0264-9381/17/18/309 (http://dx.doi.org/10.1088/0264-9381/17/18/309).fr:nilpotent

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Categories: Ring theory