In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that xn = 0.

The term was introduced by Benjamin Peirce in the context of his work on the classification of algebras.[1]

An n-by-n matrix A with entries from a field is nilpotent if and only if its characteristic polynomial is tn.

More generally, the sum of a unit element and a nilpotent element is a unit when they commute.

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.

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.[4][5] More generally, in view of the above definitions, an operator Q is nilpotent if there is nN 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,[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 used to derive theorems makes use of nilpotent or nilsquare infinitesimals, and is part smooth infinitesimal analysis.