commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...

, a Zariski ring is a commutative

Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite leng ...

topological ring In mathematics, a topological ring is a ring R that is also a topological space such that both the addition and the multiplication are continuous as maps: R \times R \to R where R \times R carries the product topology. That means R is an additive ...

''A'' whose topology is defined by an ideal

\mathfrak a

contained in the

Jacobson radical In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R- modules. It happens that substituting "left" in place of "right" in the definitio ...

, 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

is faithfully flat over ''A'' (in general, it is only flat over ''A''). * Every maximal ideal is closed.

References

* * * * Commutative algebra {{commutative-algebra-stub