Editing Integral domain (section)

== Algebraic geometry ==

Integral domains are characterized by the condition that they are [[reduced ring|reduced]] (that is {{nowrap|1=''x''<sup>2</sup> = 0}} implies {{nowrap|1=''x'' = 0}}) and [[irreducible ring|irreducible]] (that is there is only one [[minimal prime ideal]]). The former condition ensures that the [[nilradical of a ring|nilradical]] of the ring is zero, so that the intersection of all the ring's minimal primes is zero. The latter condition is that the ring have only one minimal prime. It follows that the unique minimal prime ideal of a reduced and irreducible ring is the zero ideal, so such rings are integral domains. The converse is clear: an integral domain has no nonzero nilpotent elements, and the zero ideal is the unique minimal prime ideal.

This translates, in [[algebraic geometry]], into the fact that the [[coordinate ring]] of an [[affine algebraic set]] is an integral domain if and only if the algebraic set is an [[algebraic variety]].

More generally, a commutative ring is an integral domain if and only if its [[spectrum of a ring|spectrum]] is an [[integral scheme|integral]] [[affine scheme]].