{"title":"CHARACTERIZING S-PROJECTIVE MODULES AND S-SEMISIMPLE RINGS BY UNIFORMITY","authors":"Xiaolei Zhang, W. Qi","doi":"10.1216/jca.2023.15.139","DOIUrl":null,"url":null,"abstract":"Let $R$ be a ring and $S$ a multiplicative subset of $R$. An $R$-module $P$ is called uniformly $S$-projective provided that the induced sequence $0\\rightarrow \\mathrm{Hom}_R(P,A)\\rightarrow \\mathrm{Hom}_R(P,B)\\rightarrow \\mathrm{Hom}_R(P,C)\\rightarrow 0$ is $u$-$S$-exact for any $u$-$S$-short exact sequence $0\\rightarrow A\\rightarrow B\\rightarrow C\\rightarrow 0$. Some characterizations and properties of $u$-$S$-projective modules are obtained. The notion of $u$-$S$-semisimple modules is also introduced. A ring $R$ is called a $u$-$S$-semisimple ring provided that any free $R$-module is $u$-$S$-semisimple. Several characterizations of $u$-$S$-semisimple rings are provided in terms of $u$-$S$-semisimple modules, $u$-$S$-projective modules, $u$-$S$-injective modules and $u$-$S$-split $u$-$S$-exact sequences.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"271 1","pages":""},"PeriodicalIF":0.3000,"publicationDate":"2021-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"6","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Commutative Algebra","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1216/jca.2023.15.139","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 6
Abstract
Let $R$ be a ring and $S$ a multiplicative subset of $R$. An $R$-module $P$ is called uniformly $S$-projective provided that the induced sequence $0\rightarrow \mathrm{Hom}_R(P,A)\rightarrow \mathrm{Hom}_R(P,B)\rightarrow \mathrm{Hom}_R(P,C)\rightarrow 0$ is $u$-$S$-exact for any $u$-$S$-short exact sequence $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$. Some characterizations and properties of $u$-$S$-projective modules are obtained. The notion of $u$-$S$-semisimple modules is also introduced. A ring $R$ is called a $u$-$S$-semisimple ring provided that any free $R$-module is $u$-$S$-semisimple. Several characterizations of $u$-$S$-semisimple rings are provided in terms of $u$-$S$-semisimple modules, $u$-$S$-projective modules, $u$-$S$-injective modules and $u$-$S$-split $u$-$S$-exact sequences.
期刊介绍:
Journal of Commutative Algebra publishes significant results in the area of commutative algebra and closely related fields including algebraic number theory, algebraic geometry, representation theory, semigroups and monoids.
The journal also publishes substantial expository/survey papers as well as conference proceedings. Any person interested in editing such a proceeding should contact one of the managing editors.