## Zur mengentheoretischen Darstellung gewisser Primideale. (Set theoretical representation of certain prime ideals).(German)Zbl 0589.13015

An ideal $${\mathfrak a}$$ of a commutative Noetherian ring A is said to be a set-theoretic complete intersection if there exist $$s=\dim A-\dim A/{\mathfrak a}$$ elements $$a_ 1,...,a_ s$$ such that $$\sqrt{{\mathfrak a}}=\sqrt{(a_ 1,...,a_ s)}$$ (definition 3.1).
The purpose of the paper is to give a new proof of the following theorem: Let (A,$${\mathfrak m})$$ be a local (Noetherian) ring with infinite residue field A/$${\mathfrak m}$$, and suppose the equality height $${\mathfrak a}+\dim A/{\mathfrak a}=\dim A$$ holds for every ideal $${\mathfrak a}$$. Let $${\mathfrak p}$$ be a prime ideal of height $${\mathfrak p}>0$$. If there exist elements $$x_ 1,...,x_ r$$ of $${\mathfrak m}$$ such that $$x_ 1,...,x_ r$$ is a system of parameters for A/$${\mathfrak p}$$ and $$e_ 0(({\mathfrak p},x_ 1,...,x_ r),A)=e_ 0(({\mathfrak p},x_ 1,...,x_ r)/{\mathfrak p},A/{\mathfrak p})\cdot e_ 0({\mathfrak p}A_{{\mathfrak p}},A_{{\mathfrak p}}),$$ then $${\mathfrak p}$$ is a set-theoretic complete intersection [cf. R. Achilles and W. Vogel, Math. Nachr. 89, 285-298 (1979; Zbl 0416.13015)]. The proof is done by constructing elements which possess special properties with respect to associated graded rings, superficiality etc.
Reviewer: Y.Aoyama

### MSC:

 13A15 Ideals and multiplicative ideal theory in commutative rings 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) 13H15 Multiplicity theory and related topics 14M10 Complete intersections

Zbl 0416.13015
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### References:

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