Noetherian ring in algebra
In algebra, specifically in the theory of commutative rings, a quasi-unmixed ring (also called a formally equidimensional ring in EGA) is a Noetherian ring formula_1 such that for each prime ideal "p", the completion of the localization "Ap" is equidimensional, i.e. for each minimal prime ideal "q" in the completion formula_2, formula_3 = the Krull dimension of "Ap".
Equivalent conditions.
A Noetherian integral domain is quasi-unmixed if and only if it satisfies Nagata's altitude formula. (See also: #formally catenary ring below.)
Precisely, a quasi-unmixed ring is a ring in which the unmixed theorem, which characterizes a Cohen–Macaulay ring, holds for integral closure of an ideal; specifically, for a Noetherian ring formula_1, the following are equivalent:
Formally catenary ring.
A Noetherian local ring formula_1 is said to be formally catenary if for every prime ideal formula_9, formula_10 is quasi-unmixed. As it turns out, this notion is redundant: Ratliff has shown that a Noetherian local ring is formally catenary if and only if it is universally catenary.
References.
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