# Zariski ring

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.