Krull dimension of a power series ring over a nondiscrete valuation domain is uncountable
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- Title
- Krull dimension of a power series ring over a nondiscrete valuation domain is uncountable
- Authors
- Kang, BG; Park, MH
- Date Issued
- 2013-03-15
- Publisher
- academic press inc elsevier science
- 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 Prufer domain, while the ring of entire functions is a Bezout domain. These are answers to Eakin and Sathaye's questions. Applying the above results, we show that the Krull-dimension of V[X] is uncountable if V is a nondiscrete valuation domain. (C) 2012 Elsevier Inc. All rights reserved.
- Keywords
- Commutative ring theory; Krull-dimension; Power series ring; Valuation ring
- URI
- https://oasis.postech.ac.kr/handle/2014.oak/16052
- DOI
- 10.1016/J.JALGEBRA.2012.05.017
- ISSN
- 0021-8693
- Article Type
- Article
- Citation
- journal of algebra, vol. 378, page. 12 - 21, 2013-03-15
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- There are no files associated with this item.
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