Zariski ring [SEP] named after
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
of a real closed ring to its Zariski spectrum, which sends an ordering "P" to its support formula_14 is a homeomorphism. In particular, the Zariski spectrum of every real closed ring "A" is a root system (in the sense of graph theory) and therefore "A" is also a Gel'fand ring (i.e. every prime ideal of "A" is contained in a unique maximal ideal of "A"). The comparison of the Zariski spectrum of "A" with the Zariski spectrum of "H(A