# Quasi-unmixed ring

In algebra, specifically in the theory of commutative rings, a **quasi-unmixed ring** (also called a **formally equidimensional ring** in EGA[1]) is a Noetherian ring such that for each prime ideal *p*, the completion of the localization *A _{p}* is equidimensional, i.e. for each minimal prime ideal

*q*in the completion , = the Krull dimension of

*A*.[2]

_{p}## Equivalent conditions

A Noetherian integral domain is quasi-unmixed if and only if it satisfies Nagata's altitude formula.[3] (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 , the following are equivalent:[4][5]

- is quasi-unmixed.
- For each ideal
*I*generated by a number of elements equal to its height, the integral closure 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, is unmixed.

## Formally catenary ring

A Noetherian local ring is said to be **formally catenary** if for every prime ideal , is quasi-unmixed.[6] 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.[7]

## References

- EGA IV. Part 2, Definition 7.1.1.
- Ratliff 1974, Definition 2.9. NB: "depth" there means dimension
- Ratliff 1974, Remark 2.10.1.
- Ratliff 1974, Theorem 2.29.
- Ratliff 1974, Remark 2.30.
- EGA IV. Part 2, Definition 7.1.9.
- L. J. Ratliff, Jr., Characterizations of catenary rings, Amer. J. Math. 93 (1971)

- Grothendieck, Alexandre; Dieudonné, Jean (1965). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Seconde partie".
*Publications Mathématiques de l'IHÉS*.**24**. doi:10.1007/bf02684322. MR 0199181. - Appendix of Stephen McAdam, Asymptotic Prime Divisors. Lecture notes in Mathematics.
- L.J Ratliff Jr., Locally quasi-unmixed Noetherian rings and ideals of the principal class Pacific J. Math., 52 (1974), pp. 185–205

## Further reading

- Herrmann, M., S. Ikeda, and U. Orbanz: Equimultiplicity and Blowing Up. An Algebraic Study with an Appendix by B. Moonen. Springer Verlag, Berlin Heidelberg New-York, 1988.