{"title":"Global Krull Dimension and Global Dual Krull Dimension of Valuation Rings","authors":"T. Albu, P. Vámos","doi":"10.1201/9780429187605-5","DOIUrl":"https://doi.org/10.1201/9780429187605-5","url":null,"abstract":"","PeriodicalId":139517,"journal":{"name":"abelian groups, module theory, and topology","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123403696","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-05-16DOI: 10.1201/9780429187605-23
W. Heinzer, C. Rotthaus, S. Wiegand
Suppose a is a nonzero nonunit of a Noetherian integral domain R. An interesting construction introduced by Ray Heitmann addresses the question of how ring-theoretically to adjoin a transcendental power series in a to the ring R. We apply this construction, and its natural generalization to finitely many elements, to exhibit Noetherian extension domains of R inside the (a)-adic completion R∗ of R. Suppose τ1, . . . , τs ∈ aR∗ are algebraically independent over K, the field of fractions of R. Starting with U0 := R[τ1, . . . , τs], there is a natural sequence of nested polynomial rings Un between R and A := K(τ1, . . . , τs) ∩ R∗. It is not hard to show that if U := ∪n=0Un is Noetherian, then A is a localization of U and R∗[1/a] is flat over U0. We prove, conversely, that if R∗[1/a] is flat over U0, then U is Noetherian and A := K(τ1, . . . , τs) ∩ R∗ is a localization of U . Thus the flatness of R∗[1/a] over U0 implies the intersection domain A is Noetherian.
{"title":"Building Noetherian Domains Inside an Ideal-adic Completion","authors":"W. Heinzer, C. Rotthaus, S. Wiegand","doi":"10.1201/9780429187605-23","DOIUrl":"https://doi.org/10.1201/9780429187605-23","url":null,"abstract":"Suppose a is a nonzero nonunit of a Noetherian integral domain R. An interesting construction introduced by Ray Heitmann addresses the question of how ring-theoretically to adjoin a transcendental power series in a to the ring R. We apply this construction, and its natural generalization to finitely many elements, to exhibit Noetherian extension domains of R inside the (a)-adic completion R∗ of R. Suppose τ1, . . . , τs ∈ aR∗ are algebraically independent over K, the field of fractions of R. Starting with U0 := R[τ1, . . . , τs], there is a natural sequence of nested polynomial rings Un between R and A := K(τ1, . . . , τs) ∩ R∗. It is not hard to show that if U := ∪n=0Un is Noetherian, then A is a localization of U and R∗[1/a] is flat over U0. We prove, conversely, that if R∗[1/a] is flat over U0, then U is Noetherian and A := K(τ1, . . . , τs) ∩ R∗ is a localization of U . Thus the flatness of R∗[1/a] over U0 implies the intersection domain A is Noetherian.","PeriodicalId":139517,"journal":{"name":"abelian groups, module theory, and topology","volume":"75 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126168212","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"K0 of Semiartinian Unit-Regular Rings","authors":"G. Baccella, A. Ciampella","doi":"10.1201/9780429187605-8","DOIUrl":"https://doi.org/10.1201/9780429187605-8","url":null,"abstract":"","PeriodicalId":139517,"journal":{"name":"abelian groups, module theory, and topology","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127575063","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-05-16DOI: 10.1201/9780429187605-22
J. G. Pardo, P. A. G. Asensio
{"title":"Torsionless Modules and Rings with Finite Essential Socle","authors":"J. G. Pardo, P. A. G. Asensio","doi":"10.1201/9780429187605-22","DOIUrl":"https://doi.org/10.1201/9780429187605-22","url":null,"abstract":"","PeriodicalId":139517,"journal":{"name":"abelian groups, module theory, and topology","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133218852","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"AB-5* for Module and Ring Extensions","authors":"P. N. Ánh, Dolors Herbera, C. Menini","doi":"10.1201/9780429187605-7","DOIUrl":"https://doi.org/10.1201/9780429187605-7","url":null,"abstract":"","PeriodicalId":139517,"journal":{"name":"abelian groups, module theory, and topology","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121357241","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-05-16DOI: 10.1201/9780429187605-27
S. Morris, Carolyn E. McPhail
{"title":"The Variety of Topological Groups Generated by the Class of All Banach Spaces","authors":"S. Morris, Carolyn E. McPhail","doi":"10.1201/9780429187605-27","DOIUrl":"https://doi.org/10.1201/9780429187605-27","url":null,"abstract":"","PeriodicalId":139517,"journal":{"name":"abelian groups, module theory, and topology","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131594343","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-05-16DOI: 10.1201/9780429187605-18
L. Fuchs, K. Hofmann
{"title":"Extensions of Compact Abelian Groups by Discrete Ones and Their Duality Theory, II","authors":"L. Fuchs, K. Hofmann","doi":"10.1201/9780429187605-18","DOIUrl":"https://doi.org/10.1201/9780429187605-18","url":null,"abstract":"","PeriodicalId":139517,"journal":{"name":"abelian groups, module theory, and topology","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121532128","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Lüroth’s Theorem for Some Complete Valued Fields","authors":"V. Alexandru, N. Popescu","doi":"10.1201/9780429187605-6","DOIUrl":"https://doi.org/10.1201/9780429187605-6","url":null,"abstract":"","PeriodicalId":139517,"journal":{"name":"abelian groups, module theory, and topology","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115376781","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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