Nilpotent


In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index, such that xn = 0.
The term was introduced by Benjamin Peirce in the context of his work on the classification of algebras.

Examples

No nilpotent element can be a unit. 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
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 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. This follows from the fact that nilradical is the intersection of all prime ideals.

Nilpotent elements in Lie algebra

Let be a Lie algebra. Then an element of is called nilpotent if it is in and is a nilpotent transformation. See also: Jordan decomposition in a Lie algebra.

Nilpotency in physics

An operand Q that satisfies 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 such that Qn = 0. Thus, a linear map is nilpotent iff it has a nilpotent matrix in some basis. Another example for this is the exterior derivative. 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.

Algebraic nilpotents

The two-dimensional dual numbers contain a nilpotent space. Other algebras and numbers that contain nilpotent spaces include split-quaternions, split-octonions,
biquaternions, and complex octonions .