There are exactly four absolute valued algebras of dimension four with central square roots of unit among which H, ∗ H . We present them and specify their nonzero idempotents. Mathematics Subject Classification: 17A80, 17A75, 17A60, 17D99
{"title":"Four dimensional absolute valued algebras with central square roots of unit","authors":"K. Diaby, O. Diankha, Amar Fall, A. Rochdi","doi":"10.12988/IJA.2021.91515","DOIUrl":"https://doi.org/10.12988/IJA.2021.91515","url":null,"abstract":"There are exactly four absolute valued algebras of dimension four with central square roots of unit among which H, ∗ H . We present them and specify their nonzero idempotents. Mathematics Subject Classification: 17A80, 17A75, 17A60, 17D99","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83510768","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}
Let E∓3p be elliptic curves y 2 = x ∓ 3px with prime p then, we will compute the ranks of it and compare the results with that of in curve E2pq: y 2 = x + 2pqx . Mathematics Subject Classification: 11G05, 11A41
设E - 3p为带素数p的椭圆曲线y2 = x - 3px，计算其秩，并与曲线E2pq: y2 = x + 2pqx的结果进行比较。数学学科分类:11G05, 11A41
{"title":"Ranks of several elliptic curves y2 = x3 ∓ 3px","authors":"Shin-wook Kim","doi":"10.12988/ija.2021.91565","DOIUrl":"https://doi.org/10.12988/ija.2021.91565","url":null,"abstract":"Let E∓3p be elliptic curves y 2 = x ∓ 3px with prime p then, we will compute the ranks of it and compare the results with that of in curve E2pq: y 2 = x + 2pqx . Mathematics Subject Classification: 11G05, 11A41","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77577598","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}
New simpler statements of the known theorems of Frobenius and Zorn via identities of the form (x2, y2, z2) = 0 and (x2, y2, y2) = (y2, y2, x2) = 0 are given. Also, the 124 B. Aharmim, O. Fayz, E. Idnarour and A. Rochdi identity (x2, x2, x2) = 0 in a real division algebra with a non-zero central element forces the power-commutativity. Mathematics Subject Classification: 17A35
通过(x2, y2, z2) = 0和(x2, y2, y2) = (y2, y2) = 0的恒等式给出了已知的Frobenius和Zorn定理的新的更简单的表述。此外，124 B. Aharmim, O. Fayz, E. Idnarour和a . Rochdi恒等式(x2, x2, x2) = 0的中心元非零实数除法代数强制幂交换性。数学学科分类:17A35
{"title":"Real division algebras satisfying some identities","authors":"Bouchra Aharmim, O. Fayz, E. Idnarour, A. Rochdi","doi":"10.12988/IJA.2021.91545","DOIUrl":"https://doi.org/10.12988/IJA.2021.91545","url":null,"abstract":"New simpler statements of the known theorems of Frobenius and Zorn via identities of the form (x2, y2, z2) = 0 and (x2, y2, y2) = (y2, y2, x2) = 0 are given. Also, the 124 B. Aharmim, O. Fayz, E. Idnarour and A. Rochdi identity (x2, x2, x2) = 0 in a real division algebra with a non-zero central element forces the power-commutativity. Mathematics Subject Classification: 17A35","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83778055","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}
Let A be a real division algebra with unit-element. We show that if A satisfies (x2, y2, x2) = 0 then it is flexible, quadratic and isomorphic to either R, C, a mutation of the quaternion algebra H, or a vector isotope of the octonion algebra O. Mathematics Subject Classification: 17A35 12 Kandé Diaby, Oumar Diankha, Mamoudou Ly and Abdellatif Rochdi
{"title":"Unital real division algebras satisfying (x2, y2, x2) = 0","authors":"K. Diaby, O. Diankha, M. Ly, A. Rochdi","doi":"10.12988/IJA.2021.91501","DOIUrl":"https://doi.org/10.12988/IJA.2021.91501","url":null,"abstract":"Let A be a real division algebra with unit-element. We show that if A satisfies (x2, y2, x2) = 0 then it is flexible, quadratic and isomorphic to either R, C, a mutation of the quaternion algebra H, or a vector isotope of the octonion algebra O. Mathematics Subject Classification: 17A35 12 Kandé Diaby, Oumar Diankha, Mamoudou Ly and Abdellatif Rochdi","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87895906","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}
Let A be a Banach ∗-algebra over C. In this manuscript, we study the behaviour of linear derivations with regular involution which satisfy certain differential identitities. In fact, we prove that there is no positive integer n such that the set of a ∈ A for which (a∆)n((a∗)∆)n ± ((a∗)∆)n(a∆)n ∈ Z(A ) or there exists a central idempotent e ∈ Q such that ∆ = 0 on eQ and (1 − e)Q satisfies s4, the standard identity in four variables. Mathematics Subject Classification: 16W25, 46J45
{"title":"Dense subsets on Banach *-algebras with linear derivations","authors":"H. Alhazmi","doi":"10.12988/ija.2021.91573","DOIUrl":"https://doi.org/10.12988/ija.2021.91573","url":null,"abstract":"Let A be a Banach ∗-algebra over C. In this manuscript, we study the behaviour of linear derivations with regular involution which satisfy certain differential identitities. In fact, we prove that there is no positive integer n such that the set of a ∈ A for which (a∆)n((a∗)∆)n ± ((a∗)∆)n(a∆)n ∈ Z(A ) or there exists a central idempotent e ∈ Q such that ∆ = 0 on eQ and (1 − e)Q satisfies s4, the standard identity in four variables. Mathematics Subject Classification: 16W25, 46J45","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73050494","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}
E. Salinas-Hernández, G. Ares de Parga, Jesus A EMartinez-Nuno
In this work we propose a mechanism that shortens the manual calculation of the cofactors method for the case of the determinants of the 4 × 4 and 5 × 5 matrices by means of an extension of the wellknown Sarrus’ rule. Later we give an illustrative example of the method for each case and at the end we present conclusions. Mathematics Subject Classification: 15A15,15A23,15A99
{"title":"Sarrus rule extension for 4x4 and 5x5 determinants","authors":"E. Salinas-Hernández, G. Ares de Parga, Jesus A EMartinez-Nuno","doi":"10.12988/ija.2021.91645","DOIUrl":"https://doi.org/10.12988/ija.2021.91645","url":null,"abstract":"In this work we propose a mechanism that shortens the manual calculation of the cofactors method for the case of the determinants of the 4 × 4 and 5 × 5 matrices by means of an extension of the wellknown Sarrus’ rule. Later we give an illustrative example of the method for each case and at the end we present conclusions. Mathematics Subject Classification: 15A15,15A23,15A99","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77502688","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":"Some characterizations of ternary semigroups by bi-ideals","authors":"Anila Peposhi, Thanas Xhillari","doi":"10.12988/IJA.2021.91505","DOIUrl":"https://doi.org/10.12988/IJA.2021.91505","url":null,"abstract":"","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85627199","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}
Section 17 of the important textbook [1] of M. Aschbacher studies Finite Group 1-Cohomology with a field coefficient ring via semi-direct products. This approach yields new structures and results to this basic subject. Here we assume that the coefficient ring is any commutative ring and we obtain all of the results of [1, Section 17] excluding Theorem 17.12. Via duality, this theorem extends the previous main result Theorem 17.11. In our final main results we assume that the coefficient ring is a discrete valuation ring, so that [1, Theorem 17.12] is a special case. Thus all of our results are applicable to Finite Group Modular Representation Theory. We conclude with applications to finite group permutation modules. Mathematics Subject Classification: 20J06
M. Aschbacher的重要教科书[1]第17节通过半直积研究了具有场系数环的有限群1-上同。这种方法为这一基础学科提供了新的结构和结果。这里我们假设系数环是任意可交换环，我们得到了除定理17.12外的[1,Section 17]的所有结果。通过对偶性，这个定理扩展了前面的主要结果定理17.11。在我们最后的主要结果中，我们假设系数环是一个离散估值环，因此[1，定理17.12]是一个特例。因此，我们所有的结果都适用于有限群模表示理论。最后给出了有限群置换模的应用。数学学科分类:20J06
{"title":"A note on finite group 1-cohomology via semi-direct products with applications to permutation modules","authors":"M. E. Harris","doi":"10.12988/ija.2021.91566","DOIUrl":"https://doi.org/10.12988/ija.2021.91566","url":null,"abstract":"Section 17 of the important textbook [1] of M. Aschbacher studies Finite Group 1-Cohomology with a field coefficient ring via semi-direct products. This approach yields new structures and results to this basic subject. Here we assume that the coefficient ring is any commutative ring and we obtain all of the results of [1, Section 17] excluding Theorem 17.12. Via duality, this theorem extends the previous main result Theorem 17.11. In our final main results we assume that the coefficient ring is a discrete valuation ring, so that [1, Theorem 17.12] is a special case. Thus all of our results are applicable to Finite Group Modular Representation Theory. We conclude with applications to finite group permutation modules. Mathematics Subject Classification: 20J06","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89898993","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 show that every real division algebra, with a non-zero central idempotent, satisfying (x2, y2, x2) = 0 is flexible and isomorphic to either a commutative division algebra of dimension ≤ 2, a scalar isotope of a mutation of the quaternion algebra H or a kind of isotope of the octonion algebra O. 70 Bouchra Aharmim, Kandé Diaby, Oussama Fayz and Abdellatif Rochdi Mathematics Subject Classification: 17A35
{"title":"Real division algebras with central idempotent satisfying (x^2, y^2, x^2) = 0","authors":"Bouchra Aharmim, K. Diaby, O. Fayz, A. Rochdi","doi":"10.12988/IJA.2021.91523","DOIUrl":"https://doi.org/10.12988/IJA.2021.91523","url":null,"abstract":"We show that every real division algebra, with a non-zero central idempotent, satisfying (x2, y2, x2) = 0 is flexible and isomorphic to either a commutative division algebra of dimension ≤ 2, a scalar isotope of a mutation of the quaternion algebra H or a kind of isotope of the octonion algebra O. 70 Bouchra Aharmim, Kandé Diaby, Oussama Fayz and Abdellatif Rochdi Mathematics Subject Classification: 17A35","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88035552","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}
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