Analytically unramified ring

In algebra, an analytically unramified ring is a local ring whose completion is reduced (has no nonzero nilpotent.)

The following rings are analytically unramified:

• pseudo-geometric reduced ring.
• excellent reduced ring.

There are two classical theorems of David Rees that characterize analytically unramified rings. The first says that a noetherian local ring (R, m) is analytically unramified if and only if there are a m-primary ideal J and a sequence $n_{j}\to \infty$ such that ${\overline {J^{j}}}\subset J^{n_{j}}$, where the bar means the integral closure of an ideal. The second says that a noetherian local domain is analytically unramified if and only if, for every finitely-generated R-algebra S lying between R and the field of fractions K of R, the integral closure of S in K is a finitely generated module over S. The second follows from the first.

References

• Huneke, Craig; Swanson, Irena (2006), Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, vol. 336, Cambridge, UK: Cambridge University Press, ISBN 978-0-521-68860-4, MR 2266432