{"title":"Uniform Properties of Ideals in Rings of Restricted Power Series","authors":"Madeline Grace Barnicle","doi":"10.1017/bsl.2020.26","DOIUrl":null,"url":null,"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 \n$\\mathbb Q_{p}\\langle X\\rangle $\n , \n$\\mathbb Z_{p}\\langle X\\rangle $\n , 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","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Bulletin of Symbolic Logic","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1017/bsl.2020.26","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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