Editing Exponentiation (section)

===In a ring===
In a [[ring (mathematics)|ring]], it may occur that some nonzero elements satisfy <math>x^n=0</math> for some integer {{mvar|n}}. Such an element is said to be [[nilpotent]]. In a [[commutative ring]], the nilpotent elements form an [[ideal (ring theory)|ideal]], called the [[nilradical of a ring|nilradical]] of the ring.

If the nilradical is reduced to the [[zero ideal]] (that is, if <math>x\neq 0</math> implies <math>x^n\neq 0</math> for every positive integer {{mvar|n}}), the commutative ring is said to be [[reduced ring|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 {{mvar|I}} in a commutative ring {{mvar|R}}, the set of the elements of {{mvar|R}} that have a power in {{mvar|I}} is an ideal, called the [[radical of an ideal|radical]] of {{mvar|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]] <math>k[x_1, \ldots, x_n]</math> over a [[field (mathematics)|field]] {{mvar|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]]).