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

English

Adjective

1. Describing an element, of a ring, for which there exists some positive integer n such that xn = 0.

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

0&1&0\\ 0&0&1\\ 0&0&0\end

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

0&1\\ 0&1 \end, \;\; A_2 =\begin 0&1\\ 0&0 \end \ .

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

• Idempotence