text: In a ring In a ring, it may occur that some nonzero elements satisfy  for some integer . Such an element is said to be nilpotent. In a commutative ring, the nilpotent elements form an ideal, called the nilradical of the ring.
  If the nilradical is reduced to the zero ideal (that is, if  implies  for every positive integer ), the commutative ring is said to be reduced. Reduced rings are important in algebraic geometry, since the coordinate ring of an affine algebraic set is always a reduced ring.
  More generally, given an ideal  in a commutative ring , the set of the elements of  that have a power in  is an ideal, called the radical of . The nilradical is the radical of the zero ideal. A radical ideal is an ideal that equals its own radical. In a polynomial ring  over a field , an ideal is radical if and only if it is the set of all polynomials that are zero on an affine algebraic set (this is a consequence of Hilbert's Nullstellensatz).
  Matrices and linear operators
  If  is a square matrix, then the product of  with itself  times is called the matrix power. Also  is defined to be the identity matrix, and if  is invertible, then .