{"title":"Zariski topology on the secondary-like spectrum of a module","authors":"Saif Salam, Khaldoun Al-Zoubi","doi":"10.1515/math-2024-0005","DOIUrl":null,"url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0005_eq_001.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>ℜ</m:mi> </m:math> <jats:tex-math>\\Re </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a commutative ring with unity and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0005_eq_002.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>ℑ</m:mi> </m:math> <jats:tex-math>\\Im </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a left <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0005_eq_003.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>ℜ</m:mi> </m:math> <jats:tex-math>\\Re </jats:tex-math> </jats:alternatives> </jats:inline-formula>-module. We define the secondary-like spectrum of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0005_eq_004.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>ℑ</m:mi> </m:math> <jats:tex-math>\\Im </jats:tex-math> </jats:alternatives> </jats:inline-formula> to be the set of all secondary submodules <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0005_eq_005.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>K</m:mi> </m:math> <jats:tex-math>K</jats:tex-math> </jats:alternatives> </jats:inline-formula> of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0005_eq_006.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>ℑ</m:mi> </m:math> <jats:tex-math>\\Im </jats:tex-math> </jats:alternatives> </jats:inline-formula> such that the annihilator of the socle of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0005_eq_007.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>K</m:mi> </m:math> <jats:tex-math>K</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the radical of the annihilator of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0005_eq_008.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>K</m:mi> </m:math> <jats:tex-math>K</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and we denote it by <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0005_eq_009.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi mathvariant=\"normal\">Spec</m:mi> </m:mrow> <m:mrow> <m:mi>L</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>ℑ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{{\\rm{Spec}}}^{L}\\left(\\Im )</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In this study, we introduce a topology on <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0005_eq_010.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi mathvariant=\"normal\">Spec</m:mi> </m:mrow> <m:mrow> <m:mi>L</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>ℑ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{{\\rm{Spec}}}^{L}\\left(\\Im )</jats:tex-math> </jats:alternatives> </jats:inline-formula> having the Zariski topology on the second spectrum <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0005_eq_011.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi mathvariant=\"normal\">Spec</m:mi> </m:mrow> <m:mrow> <m:mi>s</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>ℑ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{{\\rm{Spec}}}^{s}\\left(\\Im )</jats:tex-math> </jats:alternatives> </jats:inline-formula> as a subspace topology and study several topological structures of this topology.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"13 1","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Open Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/math-2024-0005","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let ℜ\Re be a commutative ring with unity and ℑ\Im be a left ℜ\Re -module. We define the secondary-like spectrum of ℑ\Im to be the set of all secondary submodules KK of ℑ\Im such that the annihilator of the socle of KK is the radical of the annihilator of KK, and we denote it by SpecL(ℑ){{\rm{Spec}}}^{L}\left(\Im ). In this study, we introduce a topology on SpecL(ℑ){{\rm{Spec}}}^{L}\left(\Im ) having the Zariski topology on the second spectrum Specs(ℑ){{\rm{Spec}}}^{s}\left(\Im ) as a subspace topology and study several topological structures of this topology.
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Open Mathematics - formerly Central European Journal of Mathematics
Open Mathematics is a fully peer-reviewed, open access, electronic journal that publishes significant, original and relevant works in all areas of mathematics. The journal provides the readers with free, instant, and permanent access to all content worldwide; and the authors with extensive promotion of published articles, long-time preservation, language-correction services, no space constraints and immediate publication.
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