Nilpotent

Examples

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.

The ring of split-quaternions contains a cone of nilpotents.

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

( 1 − x ) ( 1 + x + x 2 + ⋯ + 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 R form an ideal N ; this is a consequence of the binomial theorem. This ideal is the nilradical of the ring. Every nilpotent element x in a commutative ring is contained in every prime ideal p of that ring, since x n = 0 ∈ p . So N is contained in the intersection of all prime ideals.

If x is not nilpotent, we are able to localize with respect to the powers of x : S = { 1 , x , x 2 , . . . } to get a non-zero ring S − 1 R . The prime ideals of the localized ring correspond exactly to those prime ideals p of R with p ∩ S = ∅ . As every non-zero commutative ring has a maximal ideal, which is prime, every non-nilpotent x is not contained in some prime ideal. Thus N 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

Let g be a Lie algebra. Then an element of g is called nilpotent if it is in [ g , g ] and 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. 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.

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 C ⊗ H , and complex octonions C ⊗ O .