# 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.

## 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 class of 3 is nilpotent because 3^{2}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*c*. An example with matrices (for^{2}=(ba)^{2}=b(ab)a=0*a,b*):

- Here .

- The ring of coquaternions contains a cone of nilpotents.

## 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}, which is the case if and only if *A*^{n} = 0.

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 *x*^{n} = 0 entails

- (1 −
*x*) (1 +*x*+*x*^{2}+ ... +*x*^{n−1}) = 1 −*x*^{n}= 1.

## Nilpotency in physics

An operator that satisfies 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 *Q*^{n}=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.

## 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 Template:Doi.de:Nilpotenz