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.



  • This definition can be applied in particular to square matrices. The matrix
A = \begin{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}
\end{pmatrix}, \;\;
B =\begin{pmatrix}
Here AB = 0, BA = B.
  • The ring of coquaternions contains a cone of nilpotents.


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.

See also


  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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  • 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

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