{"title":"Some Results on Pure Ideals and Trace Ideals of Projective Modules","authors":"Abolfazl Tarizadeh","doi":"10.1007/s40306-021-00451-0","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>R</i> be a commutative ring with the unit element. It is shown that an ideal <i>I</i> in <i>R</i> is pure if and only if Ann(<i>f</i>) + <i>I</i> = <i>R</i> for all <i>f</i> ∈ <i>I</i>. If <i>J</i> is the trace of a projective <i>R</i>-module <i>M</i>, we prove that <i>J</i> is generated by the “coordinates” of <i>M</i> and <i>J</i><i>M</i> = <i>M</i>. These lead to a few new results and alternative proofs for some known results.</p></div>","PeriodicalId":45527,"journal":{"name":"Acta Mathematica Vietnamica","volume":null,"pages":null},"PeriodicalIF":0.3000,"publicationDate":"2021-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40306-021-00451-0.pdf","citationCount":"8","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Vietnamica","FirstCategoryId":"1085","ListUrlMain":"https://link.springer.com/article/10.1007/s40306-021-00451-0","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 8
Abstract
Let R be a commutative ring with the unit element. It is shown that an ideal I in R is pure if and only if Ann(f) + I = R for all f ∈ I. If J is the trace of a projective R-module M, we prove that J is generated by the “coordinates” of M and JM = M. These lead to a few new results and alternative proofs for some known results.
期刊介绍:
Acta Mathematica Vietnamica is a peer-reviewed mathematical journal. The journal publishes original papers of high quality in all branches of Mathematics with strong focus on Algebraic Geometry and Commutative Algebra, Algebraic Topology, Complex Analysis, Dynamical Systems, Optimization and Partial Differential Equations.