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
${\displaystyle A={\begin{pmatrix}0&1&0\\0&0&1\\0&0&0\end{pmatrix}}}$
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 ab in a ring R satisfy ab = 0. Then the element c = ba is nilpotent as c2 = (ba)2 = b(ab)a = 0. An example with matrices (for ab):
${\displaystyle A={\begin{pmatrix}0&1\\0&1\end{pmatrix}},\;\;B={\begin{pmatrix}0&1\\0&0\end{pmatrix}}.}$
Here AB = 0, BA = B.

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

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

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

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

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