Nilpotent

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This article is about a type of element in a ring. For the type of group, see Nilpotent group. For the type of ideal, see Nilpotent ideal.

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[edit]

  • 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.

Properties[edit]

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.

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.\

More generally, the sum of a unit element and a nilpotent element is a unit when they commute.

Commutative rings[edit]

The nilpotent elements from a commutative ring R form an ideal \mathfrak{N}; this is a consequence of the binomial theorem. This ideal is the nilradical of the ring. Every nilpotent element x in a commutative ring is contained in every prime ideal \mathfrak{p} of that ring, since x^n=0\in \mathfrak{p}. So \mathfrak{N} is contained in the intersection of all prime ideals.

If x is not nilpotent, we are able to localize with respect to the powers of x: S=\{1,x,x^2,...\} to get a non-zero ring S^{-1}R. The prime ideals of the localized ring correspond exactly to those primes \mathfrak{p} with \mathfrak{p}\cap S=\empty.[2] As every non-zero commutative ring has a maximal ideal, which is prime, every non-nilpotent x is not contained in some prime ideal. Thus \mathfrak{N} is exactly the intersection of all prime ideals.[3]

A characteristic similar to that of Jacobson radical and annihilation of simple modules is available for nilradical: nilpotent elements of ring R are precisely those that annihilate all integral domains internal to the ring R (that is, of the form R/I for prime ideals I). This follows from the fact that nilradical is the intersection of all prime ideals.

Nilpotent elements in Lie algebra[edit]

Let \mathfrak{g} be a Lie algebra. Then an element of \mathfrak{g} is called nilpotent if it is in [\mathfrak{g}, \mathfrak{g}] and \operatorname{ad} x is a nilpotent transformation. See also: Jordan decomposition in a Lie algebra.

Nilpotency in physics[edit]

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.[4][5] 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,[6] as shown by Edward Witten in a celebrated article.[7]

The electromagnetic field of a plane wave without sources is nilpotent when it is expressed in terms of the algebra of physical space.[8]

Algebraic nilpotents[edit]

The two-dimensional dual numbers contain a nilpotent space. 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[edit]

References[edit]

  1. ^ Polcino & Sehgal (2002), p. 127.
  2. ^ Matsumura, Hideyuki (1970). "Chapter 1: Elementary Results". Commutative Algebra. W. A. Benjamin. p. 6. ISBN 978-0-805-37025-6. 
  3. ^ Atiyah, M. F.; MacDonald, I. G. (February 21, 1994). "Chapter 1: Rings and Ideals". Introduction to Commutative Algebra. Westview Press. p. 5. ISBN 978-0-201-40751-8. 
  4. ^ Peirce, B. Linear Associative Algebra. 1870.
  5. ^ Milies, César Polcino; Sehgal, Sudarshan K. An introduction to group rings. Algebras and applications, Volume 1. Springer, 2002. ISBN 978-1-4020-0238-0
  6. ^ A. Rogers, The topological particle and Morse theory, Class. Quantum Grav. 17:3703–3714,2000 doi:10.1088/0264-9381/17/18/309.
  7. ^ E Witten, Supersymmetry and Morse theory. J.Diff.Geom.17:661–692,1982.
  8. ^ Rowlands, P. Zero to Infinity: The Foundations of Physics, London, World Scientific 2007, ISBN 978-981-270-914-1