ΦA,B := {I | I is non-zero finitely generated, A ⊆ I, I + B = R} . He showed, [3, Theorem 5], that M is the smallest element in ΦA,B if and only if the following conditions are satisfied: ( i) A ⊆ M , ( ii) M + B = R and ( iii) if S is a non-zero a finitely generated ideal in R such that AM−1 ⊆ S and S + B = R, then R = S. He also proved, [3, Corollary to Theorem 4], that the smallest element always exists if R is a Dedekind domain. In this short note we generalize these two results to arithmetical rings. Recall that a ring R is called an arithmetical ring if every finitely generated ideal in R is multiplication. An ideal A in R is multiplication if for every ideal B ⊆ A, there exists an ideal C in R such that B = CA, [2]. Note that C ⊆ [B : A], and hence B = CA ⊆ [B : A]A ⊆ B,
ΦA,B:= {I | I是非零有限生成的，A≤I, I + B = R}。他[3,定理5]表明,M是最小的元素在ΦB当且仅当满足以下条件:(i)⊆M, (ii) M + B = R和S (iii)是一个非零的有限生成理想的R这样−1⊆S, S + B = R, R = S .他还证明,3,推论定理4,最小的元素总是存在如果R是一个绰金环。在这个简短的笔记中，我们将这两个结果推广到算术环。回想一下，如果R中的每一个有限生成的理想都是乘法，那么一个环R就被称为算术环。如果对于每一个理想B, R中存在一个理想C，使得B = CA，则R中的理想A为乘法，[2]。请注意，C规模规模[B: A]，故B = CA规模规模[B: A]A规模规模;
{"title":"A remark on arithmetical rings","authors":"Majid M. Ali","doi":"10.12988/ija.2021.91575","DOIUrl":"https://doi.org/10.12988/ija.2021.91575","url":null,"abstract":"ΦA,B := {I | I is non-zero finitely generated, A ⊆ I, I + B = R} . He showed, [3, Theorem 5], that M is the smallest element in ΦA,B if and only if the following conditions are satisfied: ( i) A ⊆ M , ( ii) M + B = R and ( iii) if S is a non-zero a finitely generated ideal in R such that AM−1 ⊆ S and S + B = R, then R = S. He also proved, [3, Corollary to Theorem 4], that the smallest element always exists if R is a Dedekind domain. In this short note we generalize these two results to arithmetical rings. Recall that a ring R is called an arithmetical ring if every finitely generated ideal in R is multiplication. An ideal A in R is multiplication if for every ideal B ⊆ A, there exists an ideal C in R such that B = CA, [2]. Note that C ⊆ [B : A], and hence B = CA ⊆ [B : A]A ⊆ B,","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"33 5 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78248470","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We explore the nature of the continued fraction expansion of the Hurwitz numbers H = (ae2/n + b)/(ce2/n + d), with D = |ad− bc| 6 = 0. We prove some results for determinant pk with p a prime number. Also, we conjecture families of ‘pure’ Hurwitz numbers with determinant 2. Mathematics Subject Classification: 11A55, 11J70
{"title":"Characterization of pure Hurwitz numbers","authors":"J. Rodriguez, V. Bautista-Ancona","doi":"10.12988/IJA.2021.91530","DOIUrl":"https://doi.org/10.12988/IJA.2021.91530","url":null,"abstract":"We explore the nature of the continued fraction expansion of the Hurwitz numbers H = (ae2/n + b)/(ce2/n + d), with D = |ad− bc| 6 = 0. We prove some results for determinant pk with p a prime number. Also, we conjecture families of ‘pure’ Hurwitz numbers with determinant 2. Mathematics Subject Classification: 11A55, 11J70","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"44 1","pages":"111-122"},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73528043","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-10-06DOI: 10.1142/s0218196723500522
A. Atkarskaya, A. Kanel-Belov, E. Plotkin, E. Rips
In the present paper we develop a small cancellation theory for associative algebras with a basis of invertible elements. Namely, we study quotients of a group algebra of a free group and introduce three axioms for the corresponding defining relations. We show that the obtained ring is non-trivial. Moreover, we show that this ring enjoys a global filtration that agrees with relations, find a basis of the ring as a vector space and establish the corresponding structure theorems. We also provide a revision of a concept of Grobner basis for our rings and establish a greedy algorithm for the Ideal Membership Problem.
{"title":"Group-like Small Cancellation Theory for Rings","authors":"A. Atkarskaya, A. Kanel-Belov, E. Plotkin, E. Rips","doi":"10.1142/s0218196723500522","DOIUrl":"https://doi.org/10.1142/s0218196723500522","url":null,"abstract":"In the present paper we develop a small cancellation theory for associative algebras with a basis of invertible elements. Namely, we study quotients of a group algebra of a free group and introduce three axioms for the corresponding defining relations. We show that the obtained ring is non-trivial. Moreover, we show that this ring enjoys a global filtration that agrees with relations, find a basis of the ring as a vector space and establish the corresponding structure theorems. We also provide a revision of a concept of Grobner basis for our rings and establish a greedy algorithm for the Ideal Membership Problem.","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48327510","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This study aims to consider new kind of a set is called fuzzy quasi-invex set, which is convenient to handle real world application like optimization problems. And it investigates fuzzy quasi-convex set in view of new set. Some examples are given to illustrate that Zadeh’s fuzzy quasi-convex set and classical convex set are special cases of fuzzy quasi-invex set. It is also proved that classical invex sets are fuzzy quasi-invex set. Further, some properties of fuzzy quasi-invex set are investigated and generalized in term of generalized fuzzy quasi-invex set. Furthermore, some properties of fuzzy optimizations are discussed.
{"title":"On Fuzzy Quasi-Invex Sets","authors":"Muhammad Bilal Khan, M. Noor, K. Noor","doi":"10.20454/IJAS.2020.1613","DOIUrl":"https://doi.org/10.20454/IJAS.2020.1613","url":null,"abstract":"This study aims to consider new kind of a set is called fuzzy quasi-invex set, which is convenient to handle real world application like optimization problems. And it investigates fuzzy quasi-convex set in view of new set. Some examples are given to illustrate that Zadeh’s fuzzy quasi-convex set and classical convex set are special cases of fuzzy quasi-invex set. It is also proved that classical invex sets are fuzzy quasi-invex set. Further, some properties of fuzzy quasi-invex set are investigated and generalized in term of generalized fuzzy quasi-invex set. Furthermore, some properties of fuzzy optimizations are discussed.","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"29 1","pages":"11-26"},"PeriodicalIF":0.8,"publicationDate":"2020-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77928039","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-08-18DOI: 10.1142/S0218196721500417
Deepesh Singhal
A numerical semigroup is a sub-semigroup of the natural numbers that has a finite complement. Some of the key properties of a numerical semigroup are its Frobenius number [Formula: see text], genus [Formula: see text] and type [Formula: see text]. It is known that for any numerical semigroup [Formula: see text]. Numerical semigroups with [Formula: see text] are called almost symmetric, we introduce a new property that characterizes them. We give an explicit characterization of numerical semigroups with [Formula: see text]. We show that for a fixed [Formula: see text] the number of numerical semigroups with Frobenius number [Formula: see text] and type [Formula: see text] is eventually constant for large [Formula: see text]. The number of numerical semigroups with genus [Formula: see text] and type [Formula: see text] is also eventually constant for large [Formula: see text].
{"title":"Numerical semigroups of small and large type","authors":"Deepesh Singhal","doi":"10.1142/S0218196721500417","DOIUrl":"https://doi.org/10.1142/S0218196721500417","url":null,"abstract":"A numerical semigroup is a sub-semigroup of the natural numbers that has a finite complement. Some of the key properties of a numerical semigroup are its Frobenius number [Formula: see text], genus [Formula: see text] and type [Formula: see text]. It is known that for any numerical semigroup [Formula: see text]. Numerical semigroups with [Formula: see text] are called almost symmetric, we introduce a new property that characterizes them. We give an explicit characterization of numerical semigroups with [Formula: see text]. We show that for a fixed [Formula: see text] the number of numerical semigroups with Frobenius number [Formula: see text] and type [Formula: see text] is eventually constant for large [Formula: see text]. The number of numerical semigroups with genus [Formula: see text] and type [Formula: see text] is also eventually constant for large [Formula: see text].","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"31 1","pages":"883-902"},"PeriodicalIF":0.8,"publicationDate":"2020-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63937064","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-08-07DOI: 10.1142/S0218196720500575
O. Lezama, Helbert Venegas
In this paper we compute the center of many noncommutative algebras that can be interpreted as skew PBW extensions. We show that, under some natural assumptions on the parameters that define the ex...
本文计算了许多可解释为倾斜PBW扩展的非交换代数的中心。我们证明，在一些自然假设的参数下，定义前…
{"title":"Center of skew PBW extensions","authors":"O. Lezama, Helbert Venegas","doi":"10.1142/S0218196720500575","DOIUrl":"https://doi.org/10.1142/S0218196720500575","url":null,"abstract":"In this paper we compute the center of many noncommutative algebras that can be interpreted as skew PBW extensions. We show that, under some natural assumptions on the parameters that define the ex...","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"30 1","pages":"1625-1650"},"PeriodicalIF":0.8,"publicationDate":"2020-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1142/S0218196720500575","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63937018","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Enumeraion in ranks of various elliptic curves y2 = x3 ∓ AX","authors":"Shin-wook Kim","doi":"10.12988/ija.2020.91250","DOIUrl":"https://doi.org/10.12988/ija.2020.91250","url":null,"abstract":"","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"43 1","pages":"139-162"},"PeriodicalIF":0.8,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89634832","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, topological spaces based on prime LI -ideals and on maximal LI -ideals of an MTL-algebra are constructed. It is proved that the topological spaces constructed on LI -ideals are compact T0spaces and that the topological spaces constructed on maximal LI -ideals are compact T2-spaces. These give interactions between Non-Classical Mathematical Logic and General Topology. Mathematics Subject Classifications: 03G10; 54D40; 08A72
{"title":"Prime LI-ideal spaces of MTL-algebras","authors":"Chunhui Liu, Luoshan Xu","doi":"10.12988/ija.2020.91259","DOIUrl":"https://doi.org/10.12988/ija.2020.91259","url":null,"abstract":"In this paper, topological spaces based on prime LI -ideals and on maximal LI -ideals of an MTL-algebra are constructed. It is proved that the topological spaces constructed on LI -ideals are compact T0spaces and that the topological spaces constructed on maximal LI -ideals are compact T2-spaces. These give interactions between Non-Classical Mathematical Logic and General Topology. Mathematics Subject Classifications: 03G10; 54D40; 08A72","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"56 1","pages":"213-222"},"PeriodicalIF":0.8,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85945936","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, for a fixed cardinal ν, the concept of ν-C-continuous posets is introduced. Properties and characterizations of ν-C-continuous posets are presented. The main results are: (1) The lattice of all ν-Scottclosed subsets for a poset is a ν-C-continuous lattice; (2) A complete lattice is a completely distributive lattice iff it is ν-C-continuous and ν-continuous. Mathematics Subject Classifications: 06A11; 06B35; 54C35; 54D45
{"title":"Properties and characterizations of ν-C-continuous posets","authors":"Xuxin Mao, Luoshan Xu","doi":"10.12988/ija.2020.91271","DOIUrl":"https://doi.org/10.12988/ija.2020.91271","url":null,"abstract":"In this paper, for a fixed cardinal ν, the concept of ν-C-continuous posets is introduced. Properties and characterizations of ν-C-continuous posets are presented. The main results are: (1) The lattice of all ν-Scottclosed subsets for a poset is a ν-C-continuous lattice; (2) A complete lattice is a completely distributive lattice iff it is ν-C-continuous and ν-continuous. Mathematics Subject Classifications: 06A11; 06B35; 54C35; 54D45","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"54 1","pages":"223-231"},"PeriodicalIF":0.8,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83573611","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give some necessary and sufficient conditions giving rise to the finite dimensionality of any absolute-valued algebra with involution. Mathematics Subject Classification: 17A35, 17A80
给出了任意有对合的绝对值代数的有限维性的几个充分必要条件。数学学科分类:17A35、17A80
{"title":"A brief note on absolute valued algebras with involution","authors":"O. Fayz, A. Rochdi","doi":"10.12988/ija.2020.91279","DOIUrl":"https://doi.org/10.12988/ija.2020.91279","url":null,"abstract":"We give some necessary and sufficient conditions giving rise to the finite dimensionality of any absolute-valued algebra with involution. Mathematics Subject Classification: 17A35, 17A80","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"43 1","pages":"297-301"},"PeriodicalIF":0.8,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73838179","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：