Zariski ring

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

Let ''A'' be a Noetherian topological ring with the topology defined by an ideal $\mathfrak a$. Then the following are equivalent.
* ''A'' is a Zariski ring.
* The completion $\widehat$ is faithfully flat over ''A'' (in general, it is only flat over ''A'').
* Every maximal ideal is closed.

References

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*{{Citation , last1=Zariski , first1=Oscar , author1-link=Oscar Zariski , last2=Samuel , first2=Pierre , author2-link=Pierre Samuel , title=Commutative algebra. Vol. II , publisher=Springer-Verlag , location=Berlin, New York , isbn=978-0-387-90171-8 , mr=0389876 , year=1975

Commutative algebra