“Quasi-unmixed ring”的意思、由来-开放百科全书

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Quasi-unmixed ring

释义

References
Further reading

In algebra, specifically in the theory of commutative rings, a quasi-unmixed ring is a local ring such that for each minimal prime ideal p in the completion , = to the Krull dimension of A.^[1] A ring is quasi-unmixed if it is Noetherian and the localization at each prime ideal is quasi-unmixed.

A Noetherian integral domain is quasi-unmixed if and only if it satisfies Nagata's altitude formula.^[2]

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,^[3] a Noetherian ring is quasi-unmixed if and only if for each ideal I generated by a number of elements equal to its height and for each integer n > 0, the integral closure is unmixed in height (each prime divisor has the same height as the others).

References

1. ^{{harvnb|Ratliff|1974|loc=Definition 2.9. NB: "depth" there means dimension}}
2. ^{{harvnb|Ratliff|1974|loc=Remark 2.10.1.}}
3. ^{{harvnb|Ratliff|1974|loc=Theorem 2.29.}}

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

References

Further reading