Algebraic Geometry
In algebraic geometry, integral domains correspond to irreducible varieties. From the point of view of scheme theory, they have a unique generic point, given by the zero ideal. Integral domains are also characterized by the condition that they are reduced and irreducible. The former condition ensures that the 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, hence such rings are integral domains. The converse is clear: No integral domain can have nilpotent elements, and the zero ideal is the unique minimal prime ideal.
Read more about this topic: Integral Domain
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