On the ring of locally bounded Nash meromorphic functions. (English) Zbl 0911.13007
Summary: We show that the ring of locally bounded Nash meromorphic functions on a connected \(d\)-dimensional Nash submanifold of \(\mathbb{R}^n\) is a Prüfer domain and every finitely generated ideal in this ring can be generated by \(d+1\) elements. Moreover, every finitely generated ideal can be generated by \(d\) elements if and only if the Nash manifold is noncompact.
MSC:
| 13F05 | Dedekind, Prüfer, Krull and Mori rings and their generalizations |
| 14P20 | Nash functions and manifolds |
| 13E15 | Commutative rings and modules of finite generation or presentation; number of generators |
| 13G05 | Integral domains |
References:
| [1] | DOI: 10.2307/1970486 · Zbl 0122.38603 · doi:10.2307/1970486 |
| [2] | Bochnak, Géométrie Algébrique Réelle 12 (1987) |
| [3] | DOI: 10.1007/BF01168373 · Zbl 0682.12019 · doi:10.1007/BF01168373 |
| [4] | Gilmer, Multiplicative ideal theory 12 (1972) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
