Zariski ring

In commutative algebra, a Zariski ring is a commutative Noetherian topological ring "A" whose topology is defined by an ideal formula_1 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 formula_1-adic completions of Noetherian rings.
Let "A" be a Noetherian topological ring with the topology defined by an ideal formula_1. Then the following are equivalent.