# Nilpotent

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Nilpotent

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

The term was introduced by Benjamin Peirce[1] in the context of elements of an algebra that vanish when raised to a power.

## 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 ab 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 ab):
$A = \begin{pmatrix} 0&1\\ 0&1 \end{pmatrix}, \;\; B =\begin{pmatrix} 0&1\\ 0&0 \end{pmatrix}.$
Here AB = 0, BA = B.
• The ring of coquaternions contains a cone of nilpotents.

## 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.

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 + x^2 + \cdots + x^{n-1}) = 1 - x^n = 1.\$

Further, if x is nilpotent, then 1 + x is also a unit.[2]

## Nilpotency in physics

An operand Q that satisfies Q2 = 0 is nilpotent. Grassmann numbers which allow a path integral representation for Fermionic fields are nilpotents since their squares vanish. 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.[3][4] More generally, in view of the above definitions, an operator Q is nilpotent if there is nN such that Qn = 0 (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[5], as shown by Edward Witten in a celebrated article.[6]

The electromagnetic field of a plane wave without sources is nilpotent when it is expressed in terms of the algebra of physical space.[7]

## Algebraic nilpotents

The two-dimensional dual numbers contain a nilpotent basis element. Other algebras and numbers that contain nilpotent spaces include split-quaternions (coquaternions), split-octonions, biquaternions $\mathbb C\otimes\mathbb H$, and complex octonions $\mathbb C\otimes\mathbb O$.

## References

1. ^ Polcino & Sehgal (2002), p. 127.
2. ^ see Planetmath
3. ^ Peirce, B. Linear Associative Algebra. 1870.
4. ^ Milies, César Polcino; Sehgal, Sudarshan K. An introduction to group rings. Algebras and applications, Volume 1. Springer, 2002. ISBN 9781402002380
5. ^ A. Rogers, The topological particle and Morse theory, Class. Quantum Grav. 17:3703–3714,2000 doi:10.1088/0264-9381/17/18/309.
6. ^ E Witten, Supersymmetry and Morse theory. J.Diff.Geom.17:661–692,1982.
7. ^ Rowlands, P. Zero to Infinity: The Foundations of Physics, London, World Scientific 2007, ISBN 978-981-270-914-1

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### Look at other dictionaries:

• nilpotent — ● nilpotent, nilpotente adjectif Élément nilpotent d un anneau A, élément a de A tel qu il existe un entier naturel non nul n pour lequel an = 0. ● nilpotent, nilpotente (expressions) adjectif Élément nilpotent d un anneau A, élément a de A tel… …   Encyclopédie Universelle

• nilpotent —   [zu lateinisch nihil, nil »nichts«], Mathematik: 1) Ein Element a eines Ringes heißt nilpotent, wenn es eine natürliche Zahl m gibt, sodass am = 0 ist; z. B. ist   im Ring der reellen 2 ☓ 2 Matrizen nilpotent wegen a2 = 0.   2) Ein (Rechts oder …   Universal-Lexikon

• Nilpotent —  Ne doit pas être confondu avec Groupe nilpotent. En mathématiques, un élément x d un anneau unitaire (ou même d un pseudo anneau) R est appelé nilpotent s il existe un certain nombre entier positif n tel que . Sommaire …   Wikipédia en Français

• nilpotent — adjective Etymology: Latin nil nothing + potent , potens having power more at potent Date: 1870 equal to zero when raised to some power < nilpotent matrices > …   New Collegiate Dictionary

• Nilpotent — Das Adjektiv nilpotent (von lat. nihil = nichts, potens = fähig) tritt in der Mathematik in mehreren Bedeutungen auf: Als Eigenschaft von Matrizen, siehe nilpotente Matrix Allgemeiner als Eigenschaft von Ringelementen, siehe Nilpotenz Als… …   Deutsch Wikipedia

• nilpotent — /nil poht nt/, adj. Math. equal to zero when raised to a certain power. [1865 70; NIL + POTENT] * * * …   Universalium

• nilpotent — adjective Describing an element, of a ring, for which there exists some positive integer n such that x = 0. See Also: idempotent, nilpotence, nilpotency, nilpotently, nullipotent, unipotent …   Wiktionary

• nilpotent — [nɪl pəʊt(ə)nt] adjective Mathematics becoming zero when raised to some positive integral power. Origin C19: from nil + L. potens, potent power …   English new terms dictionary

• nilpotent — nil·potent …   English syllables

• nilpotent — adjective equal to zero when raised to a certain power (Freq. 12) • Pertains to noun: ↑zero …   Useful english dictionary