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.

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 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_{1}A_{2}=0,\;A_{2}A_{1}=A_{2}.

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.

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 nN 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.

The electromagnetic field of a plane wave without sources is nilpotent when it is expressed in terms of the algebra of physical space.

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 Template:Doi.de:Nilpotenz

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