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.

Contents

Examples

A = \begin{pmatrix}
0&1&0\\
0&0&1\\
0&0&0\end{pmatrix}
is nilpotent because A3 = 0. See nilpotent matrix for more.
A = \begin{pmatrix}
0&1\\
0&1
\end{pmatrix}, \;\;
B =\begin{pmatrix}
0&1\\
0&0
\end{pmatrix}.
Here AB = 0, BA = B.

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 %2B x %2B x^2 %2B \cdots %2B 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

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