The ring R such that R[α]∩R[α-1]=R is studied by Ratliff-Mirbagheri ([5]). In [7], they call α anti-integral over R if R[α]∩R[α-1]=R. In [6], the concept of an anti-integral element over R was extended to high degree case. Related papers of birational anti-integral extensions and high degree anti-integral extensions are [1], [2] and [6]. In this paper, we study the simple ring extension A/R dividing to B/R and A/B. In particular, let A=R[α] be a, primitive extension over R (see Definition 2) and put B=R[α]∩R[α-1]. Then the following statements hold. 1) A/B is flat. 2) A/R is flat if and only if B/R is flat. We give the following definition (cf. [6]).
{"title":"Some properties of certain simple flat extensions","authors":"M. Kanemitsu, KEN-ICHI Yoshida","doi":"10.5036/MJIU.29.25","DOIUrl":"https://doi.org/10.5036/MJIU.29.25","url":null,"abstract":"The ring R such that R[α]∩R[α-1]=R is studied by Ratliff-Mirbagheri ([5]). In [7], they call α anti-integral over R if R[α]∩R[α-1]=R. In [6], the concept of an anti-integral element over R was extended to high degree case. Related papers of birational anti-integral extensions and high degree anti-integral extensions are [1], [2] and [6]. In this paper, we study the simple ring extension A/R dividing to B/R and A/B. In particular, let A=R[α] be a, primitive extension over R (see Definition 2) and put B=R[α]∩R[α-1]. Then the following statements hold. 1) A/B is flat. 2) A/R is flat if and only if B/R is flat. We give the following definition (cf. [6]).","PeriodicalId":18362,"journal":{"name":"Mathematical Journal of Ibaraki University","volume":"47 1","pages":"25-29"},"PeriodicalIF":0.0,"publicationDate":"1997-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73059912","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}
When α=β=γ=0, this result is already established by H. Brezis and L. Veron in [BV]. They proved this result by using a comparison principle and a weak maximum principle. Although the operator in this paper is degenerate at the origin, their methods still work under some modifications. In fact we first construct a suitable super-solution, and then we derive a pointwise estimate by a weak muximum principle and Kato's inequality. As a application we will deduce that if u∈(C2(B') satisfies
{"title":"Notes on removable singularities for a certain class of semilinear degenerate elliptic equations","authors":"T. Horiuchi","doi":"10.5036/MJIU.29.31","DOIUrl":"https://doi.org/10.5036/MJIU.29.31","url":null,"abstract":"When α=β=γ=0, this result is already established by H. Brezis and L. Veron in [BV]. They proved this result by using a comparison principle and a weak maximum principle. Although the operator in this paper is degenerate at the origin, their methods still work under some modifications. In fact we first construct a suitable super-solution, and then we derive a pointwise estimate by a weak muximum principle and Kato's inequality. As a application we will deduce that if u∈(C2(B') satisfies","PeriodicalId":18362,"journal":{"name":"Mathematical Journal of Ibaraki University","volume":"3 1","pages":"31-39"},"PeriodicalIF":0.0,"publicationDate":"1997-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81222877","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}
Let n ≥ 1 be an integer and let 2 be the highest power of 2 dividing n. For a prime number p = 2n` + 1 with an odd prime number `, let N be the imaginary abelian field of conductor p and degree 2` over Q. We show that for n ≤ 30, the relative class number hN of N is odd when 2 is a primitive root modulo ` except for the case where (n, `) = (27, 3) and p = 163 with the help of computer.
{"title":"Note on class number parity of an abelian field of prime conductor","authors":"S. Fujima, H. Ichimura","doi":"10.5036/MJIU.50.15","DOIUrl":"https://doi.org/10.5036/MJIU.50.15","url":null,"abstract":"Let n ≥ 1 be an integer and let 2 be the highest power of 2 dividing n. For a prime number p = 2n` + 1 with an odd prime number `, let N be the imaginary abelian field of conductor p and degree 2` over Q. We show that for n ≤ 30, the relative class number hN of N is odd when 2 is a primitive root modulo ` except for the case where (n, `) = (27, 3) and p = 163 with the help of computer.","PeriodicalId":18362,"journal":{"name":"Mathematical Journal of Ibaraki University","volume":"37 1","pages":"15-26"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74717630","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}
We study semistar operations and star operations on an integral domain and we give some new characterizations of both a semistar operation and a star operation.
研究了积分域上的半星型运算和星型运算，给出了半星型运算和星型运算的一些新的性质。
{"title":"Note on characterizations of semistar operations and star operations on an integral domain","authors":"A. Okabe","doi":"10.5036/MJIU.46.31","DOIUrl":"https://doi.org/10.5036/MJIU.46.31","url":null,"abstract":"We study semistar operations and star operations on an integral domain and we give some new characterizations of both a semistar operation and a star operation.","PeriodicalId":18362,"journal":{"name":"Mathematical Journal of Ibaraki University","volume":"12 1","pages":"31-36"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77682633","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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