Krull dimension of a power series ring over a valuation domain

JOURNAL OF ALGEBRA

卷:519

Article

Phan Thanh Toan¹  Kang, Byung Gyun² 

[1] Ton Duc Thang Univ, Fac Math & Stat, Fract Calculus Optimizat & Algebra Res Grp, Ho Chi Minh City, Vietnam

[2] Pohang Univ Sci & Technol, Dept Math, Pohang 37673, South Korea

关键词: eta(1)-set;    Infinite product of power series;    Krull dimension;    Power series ring;    Ring of entire functions;    Ultrafilter;    Valuation domain;   

DOI  :  10.1016/j.jalgebra.2018.09.019

来源: Elsevier

【 摘 要 】

Let V be a one-dimensional nondiscrete valuation domain and let V* = V \ {0}. We prove that Krull-dimV[X](V*) >= 2(aleph 1), which is an analogue of the fact that Krull-dim E >= 2(aleph 1), where E is the ring of entire functions. The lower bound 2(aleph 1) is sharp. In fact, if V is countable then, Krull-dimV[X](V*) = 2(aleph 1 )under the continuum hypothesis. We construct a chain of prime ideals in V[X] with length >= 2(aleph 1) such that each prime ideal in the chain has height >= 2(aleph 1) and contracts to {0} in V. We also show that for a finite-dimensional valuation domain W, either Krull-dimW [X] < infinity or Krull-dimW [X] >= 2(aleph 1). (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

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