{"title":"满足三诺特公理的环。","authors":"J. Gilbert, H. Butts","doi":"10.32917/HMJ/1206138646","DOIUrl":null,"url":null,"abstract":"This paper is concerned with the ideal theory of a commutative ring R (which may not have an identity). We say that R is integrally closed in its total quotient ring T (or, simply, integrally closed) provided R contains every element a e T such that a is integral over R (i, e., a + rιa~-\\ [-rn = 0 for some ri, ..., rn in R). A ring R is n-dimensional (n a, non-negative integer), or has dimension n (dim R = n), provided there exists a chain P0<Pχ< <Pn<R of prime ideals in R and there is no such chain of prime ideals with greater length. If R has no prime ideals except R, then we say that dimi? = 1 . A ring is said to have property (N) provided the following three conditions are satisfied:","PeriodicalId":17080,"journal":{"name":"Journal of science of the Hiroshima University Ser. A Mathematics, physics, chemistry","volume":"90 1","pages":"211-224"},"PeriodicalIF":0.0000,"publicationDate":"1968-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Rings Satisfying the Three Noether Axioms.\",\"authors\":\"J. Gilbert, H. Butts\",\"doi\":\"10.32917/HMJ/1206138646\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This paper is concerned with the ideal theory of a commutative ring R (which may not have an identity). We say that R is integrally closed in its total quotient ring T (or, simply, integrally closed) provided R contains every element a e T such that a is integral over R (i, e., a + rιa~-\\\\ [-rn = 0 for some ri, ..., rn in R). A ring R is n-dimensional (n a, non-negative integer), or has dimension n (dim R = n), provided there exists a chain P0<Pχ< <Pn<R of prime ideals in R and there is no such chain of prime ideals with greater length. If R has no prime ideals except R, then we say that dimi? = 1 . A ring is said to have property (N) provided the following three conditions are satisfied:\",\"PeriodicalId\":17080,\"journal\":{\"name\":\"Journal of science of the Hiroshima University Ser. A Mathematics, physics, chemistry\",\"volume\":\"90 1\",\"pages\":\"211-224\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1968-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of science of the Hiroshima University Ser. A Mathematics, physics, chemistry\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.32917/HMJ/1206138646\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of science of the Hiroshima University Ser. A Mathematics, physics, chemistry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.32917/HMJ/1206138646","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 2
摘要
本文讨论了交换环R(可能没有恒等环)的理想理论。我们说R在它的全商环T中是整闭的(或者，简单地说，是整闭的)，前提是R包含每个元素a e T，使得a对R (i, e, a + rii a~-\ [-rn = 0，对于某些ri，…一个环R是n维的(n A，非负整数)，或有n维(dim R = n)，只要存在R中的素理想链P0本文章由计算机程序翻译，如有差异，请以英文原文为准。
This paper is concerned with the ideal theory of a commutative ring R (which may not have an identity). We say that R is integrally closed in its total quotient ring T (or, simply, integrally closed) provided R contains every element a e T such that a is integral over R (i, e., a + rιa~-\ [-rn = 0 for some ri, ..., rn in R). A ring R is n-dimensional (n a, non-negative integer), or has dimension n (dim R = n), provided there exists a chain P0
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