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 splitquaternions 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 nonzero nilpotent elements are zero divisors.
An nbyn 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 nonzero ring . The prime ideals of the localized ring correspond exactly to those prime ideals of with .^{[2]} As every nonzero commutative ring has a maximal ideal, which is prime, every nonnilpotent 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]}
Algebraic nilpotents
The twodimensional dual numbers contain a nilpotent space. Other algebras and numbers that contain nilpotent spaces include splitquaternions (coquaternions), splitoctonions, 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 9780805370256.
 ^ Atiyah, M. F.; MacDonald, I. G. (February 21, 1994). "Chapter 1: Rings and Ideals". Introduction to Commutative Algebra. Westview Press. p. 5. ISBN 9780201407518.
 ^ 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 9781402002380
 ^ A. Rogers, The topological particle and Morse theory, Class. Quantum Grav. 17:3703–3714, 2000 doi:10.1088/02649381/17/18/309.
 ^ 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 9789812709141