- 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 (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):
- Here AB = 0, BA = B.
- The ring of coquaternions contains a cone of nilpotents.
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
Further, if x is nilpotent, then 1 + x is also a unit.
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 two-dimensional dual numbers contain a nilpotent basis element. Other algebras and numbers that contain nilpotent spaces include split-quaternions (coquaternions), split-octonions, biquaternions , and complex octonions .
- ^ Polcino & Sehgal (2002), p. 127.
- ^ see Planetmath
- ^ Peirce, B. Linear Associative Algebra. 1870.
- ^ Milies, César Polcino; Sehgal, Sudarshan K. An introduction to group rings. Algebras and applications, Volume 1. Springer, 2002. ISBN 9781402002380
- ^ 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
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