Nilpotents

Nilpotents are elements of a ring in which some positive power equals zero. Specifically, an element x in a ring R is nilpotent if there exists k ≥ 1 with x^k = 0. This notion is most often discussed in rings with identity, where the index of nilpotency is the smallest such k.

In a commutative ring, the collection Nil(R) of all nilpotent elements forms an ideal called the nilradical. The nilradical equals the radical of the zero ideal, denoted √(0), and it is the intersection of all prime ideals of R. Consequently, Nil(R) measures the nonreduced part of the ring and its geometric counterpart in Spec(R).

Examples include: in the ring Z/8Z, the elements 2 and 4 are nilpotent since 2^3 = 8 ≡ 0 and 4^2 = 16 ≡ 0 modulo 8; in the quotient ring k[x]/(x^n), the class of x is nilpotent with index n. For matrices, a matrix over a field is nilpotent precisely when all its eigenvalues are zero; such a matrix is similar to a strictly upper triangular matrix and its nilpotency index is at most the size of the matrix.

Several properties hold in commutative rings: the sum of two nilpotent elements is nilpotent, the product of nilpotent elements is nilpotent, and any polynomial without constant term in a nilpotent element is nilpotent. In noncommutative rings, the set of nilpotent elements need not form an ideal, so the nilradical concept is most central in commutative algebra. Nilpotents thus encode a notion of "thickness" or multiplicity beyond reduced structures.