Nilpotent

In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xⁿ = 0.

Examples

This definition can be applied in particular to square matrices. The matrix

is nilpotent because A³ = 0.

In the factor ring Z/9Z, the class of 3 is nilpotent because 3² is congruent to 0 modulo 9.

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, which is the case if and only if Aⁿ = 0.

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

(1-x) (1 + x + x² + ... + x^n-1) = 1 - xⁿ = 1.