section_path: Integer powers in algebra/In a ring
section_name: In a ring
paragraph_text: In a ring
  
  In a ring, it may occur that some nonzero elements satisfy x n = 0 {\displaystyle x^{n}=0} for some integer n. 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 x ≠ 0 {\displaystyle x\neq 0} implies x n ≠ 0 {\displaystyle x^{n}\neq 0} for every positive integer n), 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 I in a commutative ring R, the set of the elements of R that have a power in I is an ideal, called the radical of I. The nilradical is the radical of the zero ideal. A radical ideal is an ideal that equals its own radical. In a polynomial ring k {\displaystyle k} over a field k, 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).