nilpotentit

Nilpotentit, in mathematics often called nilpotent elements, are elements a of a ring or algebra for which a^n = 0 for some positive integer n. The smallest such n is called the index of nilpotency.

In a ring with unity, nilpotent elements have various roles depending on the context. In a commutative ring, the set Nil(R) of all nilpotent elements forms the nilradical: an ideal that equals the intersection of all prime ideals of R. In noncommutative rings, the collection of nilpotent elements is not generally an ideal, and the structure surrounding nilpotent elements is more subtle.

In linear algebra, a square matrix A over a field is nilpotent if A^k = 0 for some k. Then all eigenvalues are zero, and the determinant and trace are zero. Equivalently, A is similar to a strictly upper triangular matrix, i.e., its Jordan form has zeros on the diagonal. A simple example is the 2×2 matrix [[0, 1], [0, 0]], which satisfies A^2 = 0.

Basic properties include the following: if a and b are commuting nilpotent elements, then their product ab is nilpotent. For any nilpotent a, the element 1 − a is a unit, with inverse 1 + a + a^2 + … + a^{n−1}. The sum of two nilpotent elements need not be nilpotent in general.

Nilpotent elements appear in several theories, including the study of the nilradical in commutative algebra, the Jordan decomposition in linear algebra, and various aspects of Lie theory where nilpotent elements play a distinctive role.