Let R be a Noetherian domain with quotient field K and let α be an anti-integral element of degree d over R. Let β be an elemen of R(α) (resp. R(α,α-1)) such that β is an anti-integral element over R and that R(α) (resp. R(α,α-1)) is integral over R(β)). We shall investigate some properties descending from R(α) (resp. R(α,α-1)) to R(β), i. e., flatness and faithful flatness, and study the ideals J(α), J(β), J(α) and J(β). Let R be a Noetherian domain and R(X) a polynomial ring. Let α be an element of an algebraic extension field L of the quotient field K of R and let π: R(X) →R(α) be the R-algebra homomorphism, sending X to α. Let ψα(X) be the monic minimal polynomial of α over K with deg ψα(X)=d and write
{"title":"Some Relationship between Anti-Integral Extensions of Noetherian Domains","authors":"Kiyoshi Baba, S. Oda, KEN-ICHI Yoshida","doi":"10.5036/MJIU.32.63","DOIUrl":"https://doi.org/10.5036/MJIU.32.63","url":null,"abstract":"Let R be a Noetherian domain with quotient field K and let α be an anti-integral element of degree d over R. Let β be an elemen of R(α) (resp. R(α,α-1)) such that β is an anti-integral element over R and that R(α) (resp. R(α,α-1)) is integral over R(β)). We shall investigate some properties descending from R(α) (resp. R(α,α-1)) to R(β), i. e., flatness and faithful flatness, and study the ideals J(α), J(β), J(α) and J(β). Let R be a Noetherian domain and R(X) a polynomial ring. Let α be an element of an algebraic extension field L of the quotient field K of R and let π: R(X) →R(α) be the R-algebra homomorphism, sending X to α. Let ψα(X) be the monic minimal polynomial of α over K with deg ψα(X)=d and write","PeriodicalId":18362,"journal":{"name":"Mathematical Journal of Ibaraki University","volume":"128 1","pages":"63-67"},"PeriodicalIF":0.0,"publicationDate":"2000-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74752031","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":"Maps between small Hopf spaces","authors":"Tomoki Egawa, H. Ōshima","doi":"10.5036/MJIU.32.33","DOIUrl":"https://doi.org/10.5036/MJIU.32.33","url":null,"abstract":"","PeriodicalId":18362,"journal":{"name":"Mathematical Journal of Ibaraki University","volume":"63 1","pages":"33-61"},"PeriodicalIF":0.0,"publicationDate":"2000-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83073940","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}
A new analytic inequality for twice differentiable convex functions and applications for the Shannon and Renyi's entropies are given.
给出了二次可微凸函数的一个新的解析不等式及其在Shannon熵和Renyi熵上的应用。
{"title":"An Inequality for Twice Differentiable Convex Functions and Applications for the Shannon and Rényi's Entropies","authors":"S. Dragomir","doi":"10.5036/MJIU.32.19","DOIUrl":"https://doi.org/10.5036/MJIU.32.19","url":null,"abstract":"A new analytic inequality for twice differentiable convex functions and applications for the Shannon and Renyi's entropies are given.","PeriodicalId":18362,"journal":{"name":"Mathematical Journal of Ibaraki University","volume":"188 1","pages":"19-28"},"PeriodicalIF":0.0,"publicationDate":"2000-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91537182","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 R be a commutative ring, and let S be a commutative semigroup. We study a semigroup version of Karpilovsky's Problem (K, chapter 7, problem 9) concerning the unit group of a group ring. We give a preciser decomposition theorem for the unit group of a semigroup ring. This is a continuation of our (M3). Thus a submonoid S of a torsion-free abelian (additive) group is called a grading monoid (or a g-monoid). Throughout the paper we assume that S is non-zero. We consider the semigroup ring R(X;S) of S over a commutative ring R. We denote the unit group of S by H=H(S). We denote the nilradical of R by N=N(R), and let U=U(R) be the unit group of R. The group of units f=Σ ααXα of R(X;S) with Σ αα=1 is denoted by
{"title":"Note on the unit group of R[X;S], II","authors":"Ryuki Matsuda","doi":"10.5036/MJIU.34.17","DOIUrl":"https://doi.org/10.5036/MJIU.34.17","url":null,"abstract":"Let R be a commutative ring, and let S be a commutative semigroup. We study a semigroup version of Karpilovsky's Problem (K, chapter 7, problem 9) concerning the unit group of a group ring. We give a preciser decomposition theorem for the unit group of a semigroup ring. This is a continuation of our (M3). Thus a submonoid S of a torsion-free abelian (additive) group is called a grading monoid (or a g-monoid). Throughout the paper we assume that S is non-zero. We consider the semigroup ring R(X;S) of S over a commutative ring R. We denote the unit group of S by H=H(S). We denote the nilradical of R by N=N(R), and let U=U(R) be the unit group of R. The group of units f=Σ ααXα of R(X;S) with Σ αα=1 is denoted by","PeriodicalId":18362,"journal":{"name":"Mathematical Journal of Ibaraki University","volume":"32 1","pages":"17-21"},"PeriodicalIF":0.0,"publicationDate":"2000-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73390498","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}
Krull in [7] conjectured that the answer to this conjecture is true, at least for the case where F is the quotient field of D and D is completely integrally closed. Nakayama [9, 10], Ohm (cf. [5, p. 232]) and Sheldon [12] gave counter examples to the conjecture. Krull proved that the conjecture holds true for one dimensional completely integrally closed quasi-local domains [8, Satz 1]. In this paper, among other things, we will prove the following facts: we characterize one dimensional Prufer domains (Corollary 2). Based on Gilmer's result [6], we prove that if F is an extension field of the quotient field K of D, then C(D), the complete integral closure of D, is the intersection of valuation
{"title":"Note on Krull's conjecture","authors":"Habte Gebru, Ryuki Matsuda","doi":"10.5036/MJIU.31.37","DOIUrl":"https://doi.org/10.5036/MJIU.31.37","url":null,"abstract":"Krull in [7] conjectured that the answer to this conjecture is true, at least for the case where F is the quotient field of D and D is completely integrally closed. Nakayama [9, 10], Ohm (cf. [5, p. 232]) and Sheldon [12] gave counter examples to the conjecture. Krull proved that the conjecture holds true for one dimensional completely integrally closed quasi-local domains [8, Satz 1]. In this paper, among other things, we will prove the following facts: we characterize one dimensional Prufer domains (Corollary 2). Based on Gilmer's result [6], we prove that if F is an extension field of the quotient field K of D, then C(D), the complete integral closure of D, is the intersection of valuation","PeriodicalId":18362,"journal":{"name":"Mathematical Journal of Ibaraki University","volume":"74 1","pages":"37-42"},"PeriodicalIF":0.0,"publicationDate":"1999-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72982889","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 D be a Noetlerian integral domain with the integral closure D, and K the quotient field of D. The Krull-Akizuki theorem states that , if dim (D) =1, then any ring between D and K is Noetherian and its dimension is at most 1. The Mori-Nagata theorem states that D is a Krull ring for any Noetherian domain D. Moreover, Nagata proved that, if D is of dimension 2 , then D is Noetherian (cf. [N, (33.12) Theorem). In [M1] we proved the Krull-Akizuki theorem for semigroups. In [M2] we proved the Mori-Nagata theorem for semigroups . The aims of this paper are to prove the following Theorem and to answer to the following question. THEOREM. Let S be a 2-dimensional Noetherian semigroup . Then the integral closure S of S is a Noetherian semigroup.
{"title":"On 2-dimensional Noetherian semigroups and a principal ideal theorem","authors":"Kojiro Sato, Ryuiki Matsuda","doi":"10.5036/MJIU.31.29","DOIUrl":"https://doi.org/10.5036/MJIU.31.29","url":null,"abstract":"Let D be a Noetlerian integral domain with the integral closure D, and K the quotient field of D. The Krull-Akizuki theorem states that , if dim (D) =1, then any ring between D and K is Noetherian and its dimension is at most 1. The Mori-Nagata theorem states that D is a Krull ring for any Noetherian domain D. Moreover, Nagata proved that, if D is of dimension 2 , then D is Noetherian (cf. [N, (33.12) Theorem). In [M1] we proved the Krull-Akizuki theorem for semigroups. In [M2] we proved the Mori-Nagata theorem for semigroups . The aims of this paper are to prove the following Theorem and to answer to the following question. THEOREM. Let S be a 2-dimensional Noetherian semigroup . Then the integral closure S of S is a Noetherian semigroup.","PeriodicalId":18362,"journal":{"name":"Mathematical Journal of Ibaraki University","volume":"9 1","pages":"29-31"},"PeriodicalIF":0.0,"publicationDate":"1999-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86949577","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}
Notation and Convensions Throughout this paper, we use the following notation unless otherwise specified: Let R be a Noetherian domain (which is commutative and has a unit), let R[X]be a polynomial ring,let. α be an element of an algebraic extension field of the quotient field K of R and let π: R[X]→R[α] be the R-algebra homomorphism sending X to α. Let ψα(X) be the monic minimal polynomial of α over K with deg ψα(X)=d and write ψα(X)=Xd+η1Xd-1+...+ηd. Then ηi ∈ K (1≦i≦d) are uniquely determined by α. Put d=[K(α):K],
{"title":"On Conditions for Denominator Ideals to Diffuse and Conditions for Elements to Be Exclusive in Anti-Integral Extensions","authors":"S. Oda, KEN-ICHI Yoshida","doi":"10.5036/MJIU.31.21","DOIUrl":"https://doi.org/10.5036/MJIU.31.21","url":null,"abstract":"Notation and Convensions Throughout this paper, we use the following notation unless otherwise specified: Let R be a Noetherian domain (which is commutative and has a unit), let R[X]be a polynomial ring,let. α be an element of an algebraic extension field of the quotient field K of R and let π: R[X]→R[α] be the R-algebra homomorphism sending X to α. Let ψα(X) be the monic minimal polynomial of α over K with deg ψα(X)=d and write ψα(X)=Xd+η1Xd-1+...+ηd. Then ηi ∈ K (1≦i≦d) are uniquely determined by α. Put d=[K(α):K],","PeriodicalId":18362,"journal":{"name":"Mathematical Journal of Ibaraki University","volume":"8 1","pages":"21-27"},"PeriodicalIF":0.0,"publicationDate":"1999-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74554662","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}
The present paper is concerned with the neccessary and sufficient conditions in order for a matrix A=(ank) to belong to the classes (l∞:Xp), (bs:Xp) and (bυ:Xp) respectively, where 1≤p≤∞. Furthermore, we prove that A∈(bs:μ) if and olny if B∈(l∞:μ) and use this to characterise the class (bs:Xp); where A and B are dual matrices and μ is any given sequence space.
{"title":"Some Matrix Transformations Into the Cesàro Sequence Spaces of Non-absolute Type","authors":"M. Şengönül, F. Başar","doi":"10.5036/MJIU.31.13","DOIUrl":"https://doi.org/10.5036/MJIU.31.13","url":null,"abstract":"The present paper is concerned with the neccessary and sufficient conditions in order for a matrix A=(ank) to belong to the classes (l∞:Xp), (bs:Xp) and (bυ:Xp) respectively, where 1≤p≤∞. Furthermore, we prove that A∈(bs:μ) if and olny if B∈(l∞:μ) and use this to characterise the class (bs:Xp); where A and B are dual matrices and μ is any given sequence space.","PeriodicalId":18362,"journal":{"name":"Mathematical Journal of Ibaraki University","volume":"17 1","pages":"13-20"},"PeriodicalIF":0.0,"publicationDate":"1999-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75068587","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}
In the present paper we essentially deal with to determine the neccessary and sufficient conditions in order for a matrix A=(ank) to belong to the classes (Xp:bs), (Xp:fs), (X1:lp), (Xp:X1) and (lp:X1), respectively. Furthermore, we give the sufficient conditions on a matrix A=(ank) in the class (Xp:lp) for 1
{"title":"Infinite Matrices and Cesàro Sequence Spaces of Non-absolute Type","authors":"F. Başar","doi":"10.5036/MJIU.31.1","DOIUrl":"https://doi.org/10.5036/MJIU.31.1","url":null,"abstract":"In the present paper we essentially deal with to determine the neccessary and sufficient conditions in order for a matrix A=(ank) to belong to the classes (Xp:bs), (Xp:fs), (X1:lp), (Xp:X1) and (lp:X1), respectively. Furthermore, we give the sufficient conditions on a matrix A=(ank) in the class (Xp:lp) for 1<p<∞ and prove a Steinhaus type theorem concerning the disjointness of the classes (Xp:fs)r and (bs:fs). Those sequence spaces are described, below.","PeriodicalId":18362,"journal":{"name":"Mathematical Journal of Ibaraki University","volume":"23 1","pages":"1-12"},"PeriodicalIF":0.0,"publicationDate":"1999-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86253001","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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