Nilpotent
In mathematics, an element, x, of a ring, R, is called nilpotent if there exists some positive integer, n, such that x^{n} = 0.
The term was introduced by Benjamin Peirce in the context of his work on the classification of algebras.^{[1]}
Contents
Examples
- This definition can be applied in particular to square matrices. The matrix
- is nilpotent because A^{3} = 0. See nilpotent matrix for more.
- In the factor ring Z/9Z, the equivalence class of 3 is nilpotent because 3^{2} is congruent to 0 modulo 9.
- Assume that two elements a, b in a ring R satisfy ab = 0. Then the element c = ba is nilpotent as c^{2} = (ba)^{2} = b(ab)a = 0. An example with matrices (for a, b):
- Here AB = 0, BA = B.
- The ring of split-quaternions contains a cone of nilpotents.
- By definition, any element of a nilsemigroup is idempotent.
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 t^{n}.
If x is nilpotent, then 1 − x is a unit, because x^{n} = 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 prime ideals of 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 Q^{2} = 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 n ∈ N such that Q^{n} = 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]} More generally, the technique of microadditivity used to derive theorems makes use of nilpotent or nilsquare infinitesimals, and is part smooth infinitesimal analysis.
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
- ^ Polcino Milies & Sehgal (2002), An Introduction to Group Rings. p. 127.
- ^ Matsumura, Hideyuki (1970). "Chapter 1: Elementary Results". Commutative Algebra. W. A. Benjamin. p. 6. ISBN 978-0-805-37025-6.
- ^ 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.
- ^ Peirce, B. Linear Associative Algebra. 1870.
- ^ Polcino Milies, César; Sehgal, Sudarshan K. An introduction to group rings. Algebras and applications, Volume 1. Springer, 2002. ISBN 978-1-4020-0238-0
- ^ A. Rogers, The topological particle and Morse theory, Class. Quantum Grav. 17:3703–3714, 2000 doi:10.1088/0264-9381/17/18/309.
- ^ E Witten, Supersymmetry and Morse theory. J.Diff.Geom.17:661–692,1982.
- ^ Rowlands, P. Zero to Infinity: The Foundations of Physics, London, World Scientific 2007, ISBN 978-981-270-914-1