The term was introduced by Benjamin Peirce in the context of his work on the classification of algebras.
- This definition can be applied in particular to square matrices. The matrix
- 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 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 a, b):
- Here AB = 0, BA = B.
- By definition, any element of a nilsemigroup is nilpotent.
If x is nilpotent, then 1 − x is a unit, because xn = 0 entails
More generally, the sum of a unit element and a nilpotent element is a unit when they commute.
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, since . So is contained in the intersection of all prime ideals.
If is not nilpotent, we are able to localize with respect to the powers of : to get a non-zero ring . The prime ideals of the localized ring correspond exactly to those prime ideals of with . As every non-zero commutative ring has a maximal ideal, which is prime, every non-nilpotent is not contained in some prime ideal. Thus is exactly the intersection of all prime ideals.
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
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. 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, 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. More generally, the technique of microadditivity used to derive theorems makes use of nilpotent or nilsquare infinitesimals, and is part smooth infinitesimal analysis.
The two-dimensional dual numbers contain a nilpotent space. Other algebras and numbers that contain nilpotent spaces include split-quaternions (coquaternions), split-octonions, biquaternions , and complex octonions .
- Polcino Milies & Sehgal (2002), An Introduction to Group Rings. p. 127.
- Matsumura, Hideyuki (1970). "Chapter 1: Elementary Results". Commutative Algebra. W. A. Benjamin. p. 6. ISBN 978-0-805-37025-6.
- 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.
- Peirce, B. Linear Associative Algebra. 1870.
- Polcino Milies, César; Sehgal, Sudarshan K. An introduction to group rings. Algebras and applications, Volume 1. Springer, 2002. ISBN 978-1-4020-0238-0
- A. Rogers, The topological particle and Morse theory, Class. Quantum Grav. 17:3703–3714, 2000 doi:10.1088/0264-9381/17/18/309.
- E Witten, Supersymmetry and Morse theory. J.Diff.Geom.17:661–692,1982.
- Rowlands, P. Zero to Infinity: The Foundations of Physics, London, World Scientific 2007, ISBN 978-981-270-914-1