Nilpotent

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

$A = \begin{pmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0 \end{pmatrix}$

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

• In the factor ring Z/9Z, the equivalence 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):

$A = \begin{pmatrix} 0 & 1\\ 0 & 1 \end{pmatrix}, \;\; B =\begin{pmatrix} 0 & 1\\ 0 & 0 \end{pmatrix}.$

Here AB = 0, BA = B.

• 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 tⁿ.

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

$(1 - x) (1 + x + x^2 + \cdots + x^{n-1}) = 1 - x^n = 1.$

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 $R$ form an ideal $\mathfrak{N}$; this is a consequence of the binomial theorem. This ideal is the nilradical of the ring. Every nilpotent element $x$ in a commutative ring is contained in every prime ideal $\mathfrak{p}$ of that ring, since $x^n = 0\in \mathfrak{p}$. So $\mathfrak{N}$ is contained in the intersection of all prime ideals.

If $x$ is not nilpotent, we are able to localize with respect to the powers of $x$: $S=\{1,x,x^2,...\}$ to get a non-zero ring $S^{-1}R$. The prime ideals of the localized ring correspond exactly to those primes $\mathfrak{p}$ with $\mathfrak{p}\cap S=\empty$.^[2] As every non-zero commutative ring has a maximal ideal, which is prime, every non-nilpotent $x$ is not contained in some prime ideal. Thus $\mathfrak{N}$ 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 $\mathfrak{g}$ be a Lie algebra. Then an element of $\mathfrak{g}$ is called nilpotent if it is in $[\mathfrak{g}, \mathfrak{g}]$ and $\operatorname{ad} x$ is a nilpotent transformation. See also: Jordan decomposition in a Lie algebra.

Nilpotency in physics

An operand Q that satisfies Q² = 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 Qⁿ = 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 two-dimensional dual numbers contain a nilpotent space. Other algebras and numbers that contain nilpotent spaces include split-quaternions (coquaternions), split-octonions, biquaternions $\mathbb C\otimes\mathbb H$, and complex octonions $\mathbb C\otimes\mathbb O$.

See also

• Idempotent element
• Unipotent
• Reduced ring
• Nil ideal

References

1. Polcino Milies & Sehgal (2002), An Introduction to Group Rings. p. 127.
2. Matsumura, Hideyuki (1970). "Chapter 1: Elementary Results". Commutative Algebra. W. A. Benjamin. p. 6. ISBN 978-0-805-37025-6.
3. 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.
4. Peirce, B. Linear Associative Algebra. 1870.
5. Polcino Milies, César; Sehgal, Sudarshan K. An introduction to group rings. Algebras and applications, Volume 1. Springer, 2002. ISBN 978-1-4020-0238-0
6. A. Rogers, The topological particle and Morse theory, Class. Quantum Grav. 17:3703–3714,2000 doi:10.1088/0264-9381/17/18/309.
7. E Witten, Supersymmetry and Morse theory. J.Diff.Geom.17:661–692,1982.
8. Rowlands, P. Zero to Infinity: The Foundations of Physics, London, World Scientific 2007, ISBN 978-981-270-914-1