Zariski ring

In commutative algebra, a Zariski ring is a commutative Noetherian topological ring A whose topology is defined by an ideal m contained in the Jacobson radical, the intersection of all maximal ideals. They were introduced by Oscar Zariski (1946) under the name "semi-local ring" which now means something different, and named "Zariski rings" by Samuel (1953). Examples of Zariski rings are noetherian local rings and $\mathfrak a$-adic completions of noetherian rings.

Let A be a noetherian ring and $\widehat{A}$ its $\mathfrak a$-adic completion. Then the following are equivalent.

• $\widehat{A}$ is faithfully flat over A (in general, only flat over it).
• Every maximal ideal is closed for the $\mathfrak a$-adic topology.
• A is a Zariski ring.

References

• M. Atiyah, I. Macdonald Introduction to commutative algebra Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969
• Samuel, Pierre (1953), Algèbre locale, Mémor. Sci. Math. 123, Paris: Gauthier-Villars, MR 0054995
• Zariski, Oscar (1946), "Generalized semi-local rings", Summa Brasil. Math. 1 (8): 169–195, MR 0022835
• Zariski, Oscar; Samuel, Pierre (1975), Commutative algebra. Vol. II, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90171-8, MR 0389876