Nilpotent

From formulasearchengine
Jump to navigation Jump to search

{{#invoke:Hatnote|hatnote}}

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
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):
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.

If x is nilpotent, then 1 − x is a unit, because xn = 0 entails

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

Commutative rings

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, since . So is contained in the intersection of all prime ideals.

If is not nilpotent, we are able to localize with respect to the powers of : to get a non-zero ring . The prime ideals of the localized ring correspond exactly to those primes with .[2] As every non-zero commutative ring has a maximal ideal, which is prime, every non-nilpotent is not contained in some prime ideal. Thus 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

Let be a Lie algebra. Then an element of is called nilpotent if it is in and is a nilpotent transformation. See also: Jordan decomposition in a Lie algebra.

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.[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

The two-dimensional dual numbers contain a nilpotent space. Other algebras and numbers that contain nilpotent spaces include split-quaternions (coquaternions), split-octonions, biquaternions , and complex octonions .

See also

References

  1. Polcino & Sehgal (2002), p. 127.
  2. {{#invoke:citation/CS1|citation |CitationClass=book }}
  3. {{#invoke:citation/CS1|citation |CitationClass=book }}
  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 Template:Hide in printTemplate:Only in print.
  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