In commutative algebra, a Zariski ring is a commutative Noetherian topological ring A whose topology is defined by an ideal 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 -adic completions of Noetherian rings.
Let A be a Noetherian topological ring with the topology defined by an ideal . Then the following are equivalent. A is a Zariski ring. The completion is faithfully flat over A (in general, it is only flat over A). Every maximal ideal is closed.
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