Zariski ring

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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.