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 $$\widehat{A}$$ is faithfully flat over A (in general, it is only flat over A).
 * Every maximal ideal is closed.