{"title":"半群环的Krull性质","authors":"Ryuki Matsuda","doi":"10.5036/BFSIU1968.14.1","DOIUrl":null,"url":null,"abstract":"ΣaαXα for aα∈D and α∈G almost all aα are zero. Various algebraic properties have been studied by various authors ([1],[3],[4],[5],[8],[9],[10],[11],[12], [13],[14],[15],[16] etc.). We concern Krull properties of the group ring D[X;G] in this paper. Let K be the quotient field of D and let F={Vλ;λ ∈ Λ} be a set of valuation rings of k. We concern following properties on F: (E1) D=∩{Vλ;λ ∈Λ}; (E2) Each Vλ is rank 1 discrete; (E2)' Each Vλ has rank 1; (E2)\" Each Vλ is a rational number valued valuation ring; (E3) F has finite character-that is, if 0≠x∈K, then x is a nonunit in only finitely many of the valuation rings in F;","PeriodicalId":141145,"journal":{"name":"Bulletin of The Faculty of Science, Ibaraki University. Series A, Mathematics","volume":"22 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"1982-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"8","resultStr":"{\"title\":\"Krull Properties of Semigroup Rings\",\"authors\":\"Ryuki Matsuda\",\"doi\":\"10.5036/BFSIU1968.14.1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"ΣaαXα for aα∈D and α∈G almost all aα are zero. Various algebraic properties have been studied by various authors ([1],[3],[4],[5],[8],[9],[10],[11],[12], [13],[14],[15],[16] etc.). We concern Krull properties of the group ring D[X;G] in this paper. Let K be the quotient field of D and let F={Vλ;λ ∈ Λ} be a set of valuation rings of k. We concern following properties on F: (E1) D=∩{Vλ;λ ∈Λ}; (E2) Each Vλ is rank 1 discrete; (E2)' Each Vλ has rank 1; (E2)\\\" Each Vλ is a rational number valued valuation ring; (E3) F has finite character-that is, if 0≠x∈K, then x is a nonunit in only finitely many of the valuation rings in F;\",\"PeriodicalId\":141145,\"journal\":{\"name\":\"Bulletin of The Faculty of Science, Ibaraki University. Series A, Mathematics\",\"volume\":\"22 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1982-05-31\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"8\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Bulletin of The Faculty of Science, Ibaraki University. Series A, Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.5036/BFSIU1968.14.1\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of The Faculty of Science, Ibaraki University. Series A, Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.5036/BFSIU1968.14.1","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
ΣaαXα for aα∈D and α∈G almost all aα are zero. Various algebraic properties have been studied by various authors ([1],[3],[4],[5],[8],[9],[10],[11],[12], [13],[14],[15],[16] etc.). We concern Krull properties of the group ring D[X;G] in this paper. Let K be the quotient field of D and let F={Vλ;λ ∈ Λ} be a set of valuation rings of k. We concern following properties on F: (E1) D=∩{Vλ;λ ∈Λ}; (E2) Each Vλ is rank 1 discrete; (E2)' Each Vλ has rank 1; (E2)" Each Vλ is a rational number valued valuation ring; (E3) F has finite character-that is, if 0≠x∈K, then x is a nonunit in only finitely many of the valuation rings in F;