In commutative algebra, a Zariski ring is a commutative Noetherian topological ring A whose topology is defined by an ideal ${\displaystyle {\mathfrak {a}}}$ 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 Pierre Samuel ( 1953). Examples of Zariski rings are noetherian local rings with the topology induced by the maximal ideal, and ${\displaystyle {\mathfrak {a}}}$-adic completions of Noetherian rings.

Let A be a Noetherian topological ring with the topology defined by an ideal ${\displaystyle {\mathfrak {a}}}$. Then the following are equivalent.

• A is a Zariski ring.
• The completion ${\displaystyle {\widehat {A}}}$ is faithfully flat over A (in general, it is only flat over A).
• Every maximal ideal is closed.

References

• Atiyah, Michael F.; Macdonald, Ian G. (1969), Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., MR  0242802
• Samuel, Pierre (1953), Algèbre locale, Mémor. Sci. Math., vol. 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

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