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Constructing bad Noetherian local domains using derivations. (English) Zbl 0692.13005

This paper contains an explicit construction of commutative noetherian local domains R of dimension 1 or 2 having the integral closure infinitely generated as an R-module. Such examples are constructed having arbitrary embedding dimension and multiplicity. Moreover, R supports a derivation leaving no ideals invariant. The method is to define a subring R of a power series algebra by using a derivation. The computations required to show that R has the desired properties are rather straightforward.
Reviewer: M.Cipu

MSC:

13E05 Commutative Noetherian rings and modules
13B10 Morphisms of commutative rings
13G05 Integral domains
13H99 Local rings and semilocal rings
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[1] Akizuki, Y., Einige Bemerkungenu¨ber prima¨re Integrita¨tsbereiche mit Teilerkettensatz, (Proc. Phys.-Math. Soc. Japan, 17 (1935)), 327-336 · JFM 61.1029.02
[2] de Souza Doering, A. M.; Lequain, Y., Maximally differential prime ideals, J. Algebra, 101, 403-417 (1986) · Zbl 0591.13018
[3] Ferrand, D.; Raynaud, M., Fibres formelles d’un anneau local noethe´rien, Ann. Sci.E´cole Norm. Sup., 3, 295-311 (1970), (4) · Zbl 0204.36601
[4] Lequain, Y., Differential simplicity and complete integral closure, Pacific J. Math., 36, 741-751 (1971) · Zbl 0188.09702
[5] Nagata, M., Local Rings (1962), Interscience: Interscience New York · Zbl 0123.03402
[6] Seidenberg, A., Derivations and integral closure, Pacific J. Math., 16, 167-173 (1966) · Zbl 0133.29202
[7] Vasconcelos, W. V., Derivations of commutative noetherian rings, Math. Z., 112, 229-233 (1969) · Zbl 0181.05201
[8] Zariski, O.; Samuel, P., (Commutative Algebra, Vol. I (1958), Van Nostrand: Van Nostrand Princeton, NJ) · Zbl 0112.02902
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