Riassunto
Anne MarieNicolas in [3] pose la seguente domanda: esiste un dominio di valutazioneA avente un ideale che non può essere generato da un insieme numerabile di elementi e tale che il suo campo di quozienti abbia dimensione omologica 1?
Noi rispondiamo a questa domanda affermativamente.
Summary
In [3], Anne MarieNicolas asked the following question:
Does there exist a valuation domainA, having an ideal which cannot be generated by a countable set of elements, and such that its field of quotients has homological dimension 1?
We answer this question in the affirmative.
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References
L. Fuchs,Partially Ordered Algebraic Systems, Pergamon Press, Oxford, 1963.
I. Kaplansky,The homological dimension of a quotient field, Nagoya Math. J.,27 (1966), pp. 139–142.
A. M. Nicolas,Gènéralisation d'un critère de Pontrjagin concernant les groupes sans torsion dénombrables à des modules sans torsion sur des anneaux de Dedekind. Conditions de rang, de type, de chaînes ascendantes, Springer Lecture Notes, no. 740 (1979), pp. 385–396.
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Ribenboim, P. On a question of Anne Marie Nicolas concerning valuation rings. Ann. Univ. Ferrara 28, 81–84 (1982). https://doi.org/10.1007/BF02900755
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DOI: https://doi.org/10.1007/BF02900755