Uniform Properties of Ideals in Rings of Restricted Power Series

Madeline Grace Barnicle
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Abstract

Abstract When is an ideal of a ring radical or prime? By examining its generators, one may in many cases definably and uniformly test the ideal’s properties. We seek to establish such definable formulas in rings of p-adic power series, such as $\mathbb Q_{p}\langle X\rangle $ , $\mathbb Z_{p}\langle X\rangle $ , and related rings of power series over more general valuation rings and their fraction fields. We obtain a definable, uniform test for radicality, and, in the one-dimensional case, for primality. This builds upon the techniques stemming from the proof of the quantifier elimination results for the analytic theory of the p-adic integers by Denef and van den Dries, and the linear algebra methods of Hermann and Seidenberg. Abstract prepared by Madeline G. Barnicle. E-mail: barnicle@math.ucla.edu URL: https://escholarship.org/uc/item/6t02q9s4
限制幂级数环上理想的一致性质
什么时候是环的理想基或素数?通过检验它的产生源,人们可以在许多情况下明确而一致地检验理想的性质。我们试图在p进幂级数环中建立这样的可定义公式,例如$\mathbb Q_{p}\langle X\rangle $, $\mathbb Z_{p}\langle X\rangle $,以及在更一般的赋值环及其分数域上幂级数的相关环。我们得到了一个可定义的、一致的根性检验,并在一维情况下得到了素数检验。这建立在Denef和van den Dries对p进整数解析理论的量词消去结果的证明以及Hermann和Seidenberg的线性代数方法所产生的技术之上。摘要由Madeline G. Barnicle制备。电子邮件:barnicle@math.ucla.edu URL: https://escholarship.org/uc/item/6t02q9s4
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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