is called nilpotent if there exists some positive integer
, called the index (or sometimes the degree), such that
The term, along with its sister idempotent, was introduced by Benjamin Peirce in the context of his work on the classification of algebras.
All nilpotent elements are zero divisors.
with entries from a field is nilpotent if and only if its characteristic polynomial is
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
; this is a consequence of the binomial theorem.
in a commutative ring is contained in every prime ideal
is contained in the intersection of all prime ideals.
is not nilpotent, we are able to localize with respect to the powers of
The prime ideals of the localized ring correspond exactly to those prime ideals
[2] As every non-zero commutative ring has a maximal ideal, which is prime, every non-nilpotent
is not contained in some prime ideal.
[3] A characteristic similar to that of Jacobson radical and annihilation of simple modules is available for nilradical: nilpotent elements of a ring
are precisely those that annihilate all integral domains internal to the ring
This follows from the fact that nilradical is the intersection of all prime ideals.
See also: Jordan decomposition in a Lie algebra.
Any ladder operator in a finite dimensional space is nilpotent.
They represent creation and annihilation operators, which transform from one state to another, for example the raising and lowering Pauli matrices
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
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 (which can used to derive theorems in physics) 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
If a nilpotent infinitesimal is a variable tending to zero, it can be shown that any sum of terms for which it is the subject is an indefinitely small proportion of the first order term.