# 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]

## 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**/9**Z**, 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 nilpotent.

## 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 .

## 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