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.

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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