Nilpotence

Nilpotence in algebra refers to an element a of a ring R such that a^n = 0 for some positive integer n. The least such n, if it exists, is called the nilpotency index of a. The zero element is nilpotent with index 1. Nilpotent elements are annihilated by some power, and this property is preserved under taking powers.

In matrix theory, a nilpotent matrix N satisfies N^k = 0 for some k. Over a field, such a matrix has all eigenvalues equal to zero and is similar to a strictly upper triangular matrix. The nilpotency index is at most the size of the matrix and equals the size of the largest Jordan block corresponding to the eigenvalue zero.

In commutative algebra, the set Nil(R) of all nilpotent elements in a ring R is called the nilradical. It equals the radical of the zero ideal and, equivalently, the intersection of all prime ideals of R. If R is Noetherian, Nil(R) is a nilpotent ideal, and the quotient R/Nil(R) is a reduced ring (having no nonzero nilpotent elements). In a commutative ring, Nil(R) forms an ideal and plays a key role in algebraic geometry as it describes infinitesimal thickening of varieties.

For general rings, nilpotent elements always satisfy a^n = 0, and they lie in the Jacobson radical (the intersection of maximal ideals). The nilradical may differ from the Jacobson radical in noncommutative settings, but Nil(R) is contained in J(R). Nilpotence also appears in the theory of nilpotent ideals and in linear algebra through nilpotent linear operators and their Jordan forms.