# Dictionary Definition

nilpotent adj : equal to zero when raised to a
certain power

# User Contributed Dictionary

## English

### Adjective

# Extensive Definition

In mathematics, an element x of
a ring R
is called nilpotent if there exists some positive integer n such that xn =
0.

## Examples

- This definition can be applied in particular to square matrices. The matrix

- A = \begin

- is nilpotent because A3 = 0. See nilpotent matrix for more.

- In the factor ring Z/9Z, the class of 3 is nilpotent because 32 is congruent to 0 modulo 9.

- Assume that two elements a,b 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 a,b):

- A_1 = \begin

- Here A_1A_2=0,\; A_2A_1=A_2 .

- The ring of coquaternions contains a cone of nilpotents.

## Properties

No nilpotent element can be a unit
(except in the trivial ring
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 .

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, and in fact the intersection of all these
prime ideals is equal to the nilradical.

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

- (1 − x) (1 + x + x2 + ... + xn−1) = 1 − xn = 1.

## Nilpotency in physics

An operator Q that satisfies Q^2=0
is nilpotent. 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.
More generally, in view of the above definitions, an operator Q is
nilpotent if there is n∈N such that Qn=o (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, as shown by Edward
Witten in a celebrated article.

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

## Algebraic nilpotents

The following are examples of algebras and
numbers that contain nilpotents:

- Split-quaternion / coquaternion
- Split-octonion
- Conic sedenions from Musean hypernumbers

## References

- E Witten, Supersymmetry and Morse theory. J.Diff.Geom.17:661-692,1982.
- A. Rogers, The topological particle and Morse theory, Class. Quantum Grav. 17:3703-3714,2000 .

## See also

nilpotent in Catalan: Nilpotent

nilpotent in German: Nilpotenz

nilpotent in Spanish: Nilpotente

nilpotent in French: Nilpotent

nilpotent in Hebrew: נילפוטנטיות

nilpotent in Italian: Nilpotente

nilpotent in Hungarian: Nilpotens elem

nilpotent in Dutch: Nilpotent

nilpotent in Polish: Element nilpotentny

nilpotent in Portuguese: Nilpotente

nilpotent in Russian: Нильпотентный
элемент

nilpotent in Thai: นิรพล

nilpotent in Chinese: 幂零元