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- W1975722401 abstract "Let V be a rank-one nondiscrete valuation domain with maximal ideal M . We prove that the Krull-dimension of V 〚 X 〛 V ∖ ( 0 ) is uncountable, and hence the Krull-dimension of V 〚 X 〛 is uncountable. This corresponds to the well-known fact that the Krull-dimension of the ring of entire functions is uncountable. In fact we construct an uncountable chain of prime ideals inside M 〚 X 〛 such that all the members contract to (0) in V . Our method provides a new proof that the Krull-dimension of the ring of entire functions is uncountable. It is also shown that V 〚 X 〛 V ∖ ( 0 ) is not even a Prüfer domain, while the ring of entire functions is a Bezout domain. These are answers to Eakin and Sathayeʼs questions. Applying the above results, we show that the Krull-dimension of V 〚 X 〛 is uncountable if V is a nondiscrete valuation domain." @default.
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- W1975722401 date "2013-03-01" @default.
- W1975722401 modified "2023-09-28" @default.
- W1975722401 title "Krull-dimension of the power series ring over a nondiscrete valuation domain is uncountable" @default.
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- W1975722401 doi "https://doi.org/10.1016/j.jalgebra.2012.05.017" @default.
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