{"title":"通过扭转模对射影模的无扭转扩展","authors":"L. Fuchs","doi":"10.1216/jca.2023.15.31","DOIUrl":null,"url":null,"abstract":"We consider a generalization of a problem raised by P. Griffith [12] on abelian groups to modules over integral domains, and prove an analogue of a theorem of M. Dugas and J. Irwin [2]. Torsion modules T with the following property are characterized: if M is a torsion-free module and F is a projective submodule such that M/F ∼= T , then M is projective (Theorem 4.1). It is shown in Theorem 6.4 that for abelian groups whose cardinality is not cofinal with ω this is equivalent to being totally reduced in the sense of L. Fuchs and K. Rangaswamy [9]. The problem for valuation domains is also discussed, the results are similar to the case of abelian groups.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"TORSION-FREE EXTENSIONS OF PROJECTIVE MODULES BY TORSION MODULES\",\"authors\":\"L. Fuchs\",\"doi\":\"10.1216/jca.2023.15.31\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We consider a generalization of a problem raised by P. Griffith [12] on abelian groups to modules over integral domains, and prove an analogue of a theorem of M. Dugas and J. Irwin [2]. Torsion modules T with the following property are characterized: if M is a torsion-free module and F is a projective submodule such that M/F ∼= T , then M is projective (Theorem 4.1). It is shown in Theorem 6.4 that for abelian groups whose cardinality is not cofinal with ω this is equivalent to being totally reduced in the sense of L. Fuchs and K. Rangaswamy [9]. The problem for valuation domains is also discussed, the results are similar to the case of abelian groups.\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2023-03-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1216/jca.2023.15.31\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1216/jca.2023.15.31","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
TORSION-FREE EXTENSIONS OF PROJECTIVE MODULES BY TORSION MODULES
We consider a generalization of a problem raised by P. Griffith [12] on abelian groups to modules over integral domains, and prove an analogue of a theorem of M. Dugas and J. Irwin [2]. Torsion modules T with the following property are characterized: if M is a torsion-free module and F is a projective submodule such that M/F ∼= T , then M is projective (Theorem 4.1). It is shown in Theorem 6.4 that for abelian groups whose cardinality is not cofinal with ω this is equivalent to being totally reduced in the sense of L. Fuchs and K. Rangaswamy [9]. The problem for valuation domains is also discussed, the results are similar to the case of abelian groups.