{"title":"限制幂级数环上理想的一致性质","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 $\n , and related rings of power series over more general valuation rings and their fraction fields. We obtain a definable, uniform test for radicality, and, in the one-dimensional case, for primality. This builds upon the techniques stemming from the proof of the quantifier elimination results for the analytic theory of the p-adic integers by Denef and van den Dries, and the linear algebra methods of Hermann and Seidenberg. Abstract prepared by Madeline G. Barnicle. E-mail: barnicle@math.ucla.edu URL: https://escholarship.org/uc/item/6t02q9s4","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Uniform Properties of Ideals in Rings of Restricted Power Series\",\"authors\":\"Madeline Grace Barnicle\",\"doi\":\"10.1017/bsl.2020.26\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract When is an ideal of a ring radical or prime? By examining its generators, one may in many cases definably and uniformly test the ideal’s properties. We seek to establish such definable formulas in rings of p-adic power series, such as \\n$\\\\mathbb Q_{p}\\\\langle X\\\\rangle $\\n , \\n$\\\\mathbb Z_{p}\\\\langle X\\\\rangle $\\n , and related rings of power series over more general valuation rings and their fraction fields. We obtain a definable, uniform test for radicality, and, in the one-dimensional case, for primality. This builds upon the techniques stemming from the proof of the quantifier elimination results for the analytic theory of the p-adic integers by Denef and van den Dries, and the linear algebra methods of Hermann and Seidenberg. Abstract prepared by Madeline G. Barnicle. E-mail: barnicle@math.ucla.edu URL: https://escholarship.org/uc/item/6t02q9s4\",\"PeriodicalId\":22265,\"journal\":{\"name\":\"The Bulletin of Symbolic Logic\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-06-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"The Bulletin of Symbolic Logic\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1017/bsl.2020.26\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Bulletin of Symbolic Logic","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1017/bsl.2020.26","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
什么时候是环的理想基或素数?通过检验它的产生源,人们可以在许多情况下明确而一致地检验理想的性质。我们试图在p进幂级数环中建立这样的可定义公式,例如$\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
Uniform Properties of Ideals in Rings of Restricted Power Series
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