Pairs of domains where all intermediate domains are Noetherian

Author:
Adrian R. Wadsworth

Journal:
Trans. Amer. Math. Soc. **195** (1974), 201-211

MSC:
Primary 13G05

DOI:
https://doi.org/10.1090/S0002-9947-1974-0349665-2

MathSciNet review:
0349665

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Abstract: For Noetherian integral domains *R* and *T* with is called a *Noetherian pair* (NP) if every domain *A*, , is Noetherian. When (Krull dimension) it is shown that the only NP's are those given by the Krull-Akizuki Theorem. For , there is another type of NP besides the finite integral extension, namely where . Further, for every NP (*R, T*) with there is an integral NP extension *B* of *R* with . In all known examples *B* can be chosen to be a *finite* integral extension of *R*. For such NP's it is shown that the NP relation is transitive. *T* may itself be an infinite integral extension *R*, though, and an example of this is given. It is unknown exactly which infinite integral extensions are NP's.

**[1]**N. Bourbaki,*Éléments de mathématique. Fasc. XXX. Algèbre commutative. Chapitre 5: Entiers. Chapitre 6: Valuations*, Actualités Scientifiques et Industrielles, No. 1308, Hermann, Paris, 1964 (French). MR**0194450****[2]**Edward D. Davis,*Integrally closed pairs*, Conference on Commutative Algebra (Univ. Kansas, Lawrence, Kan., 1972), Springer, Berlin, 1973, pp. 103–106. Lecture Notes in Math., Vol. 311. MR**0335490****[3]**Edward D. Davis,*Overrings of commutative rings. I. Noetherian overrings*, Trans. Amer. Math. Soc.**104**(1962), 52–61. MR**0139629**, https://doi.org/10.1090/S0002-9947-1962-0139629-8**[4]**Paul M. Eakin Jr.,*The converse to a well known theorem on Noetherian rings*, Math. Ann.**177**(1968), 278–282. MR**0225767**, https://doi.org/10.1007/BF01350720**[5]**Robert Gilmer,*Integral domains with Noetherian subrings*, Comment. Math. Helv.**45**(1970), 129–134. MR**0262216**, https://doi.org/10.1007/BF02567320**[6]**Irving Kaplansky,*Commutative rings*, Allyn and Bacon, Inc., Boston, Mass., 1970. MR**0254021****[7]**-,*Topics in commutative rings*. I, mimeographed notes.**[8]**Masayoshi Nagata,*Local rings*, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers a division of John Wiley & Sons New York-London, 1962. MR**0155856****[9]**Oscar Zariski and Pierre Samuel,*Commutative algebra. Vol. II*, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N. J.-Toronto-London-New York, 1960. MR**0120249**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1974-0349665-2

Keywords:
Finitely generated module,
integral extension,
Krull dimension,
Krull-Akizuki Theorem,
Noetherian integral domain

Article copyright:
© Copyright 1974
American Mathematical Society