Excellent ring

Summary

An excellent ring is a Noetherian ring that satisfies three extra conditions on chains of primes, on completions and on regular loci. The class was introduced by Grothendieck to collect the Noetherian rings that behave as well as the rings met in geometry and number theory. Fields, the integers, complete Noetherian local rings and all finitely generated algebras over them are excellent. For excellent rings, properties such as being reduced or normal pass from a local ring to its completion. Also, the integral closure of a reduced excellent ring is a finite module over it.