In commutative algebra

Commutative algebra

Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...

, a Zariski ring, introduced by , is a commutative Noetherian

Noetherian ring

In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...

 topological ring A whose topology is defined by an ideal m contained in the Jacobson radical

Jacobson radical

In mathematics, more specifically ring theory, a branch of abstract algebra, the Jacobson radical of a ring R is an ideal which consists of those elements in R which annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition yields the same...

, the intersection of all maximal ideals. Examples of Zariski rings are notherian local rings and -adic completions of notherian rings.

Let A be a noetherian ring and its -adic completion. Then the following are equivalent.

• is faithfully flat over A (in general, only flat over it).
• Every maximal ideal is closed for the -adic topology.
• A is a Zariski ring.