Quasi-unmixed ring

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 $$A$$ 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 $$\widehat{A_p}$$, $$\dim \widehat{A_p}/q = \dim A_p$$ = 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: 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 $$A$$, the following are equivalent:
 * $$A$$ is quasi-unmixed.
 * For each ideal I generated by a number of elements equal to its height, the integral closure $$\overline{I}$$ is unmixed in height (each prime divisor has the same height as the others).
 * For each ideal I generated by a number of elements equal to its height and for each integer n > 0, $$\overline{I^n}$$ is unmixed.

Formally catenary ring
A Noetherian local ring $$A$$ is said to be formally catenary if for every prime ideal $$\mathfrak{p}$$, $$A/\mathfrak{p}$$ 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.