Zariski ring

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

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

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

References

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