We reformulate the examinations of the Apéry set of a numerical semigroup of embedding dimension 4 with respect to a generator n to the examinations of a presentation of the cyclic group of order n , by three generators and three relations. We examine such presentations using the GAP package
{"title":"An investigation of numerical semigroups of embedding dimension 4 through GAP","authors":"Merita Bajrami, D. Dimovski, Violeta Angjelkoska","doi":"10.12988/ija.2023.91732","DOIUrl":"https://doi.org/10.12988/ija.2023.91732","url":null,"abstract":"We reformulate the examinations of the Apéry set of a numerical semigroup of embedding dimension 4 with respect to a generator n to the examinations of a presentation of the cyclic group of order n , by three generators and three relations. We examine such presentations using the GAP package","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79715930","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}
The aim of this paper is to construct, by algebraic methods, some new class of four-dimensional absolute valued algebras, namely M 1 , M 2 and M 3 which are not isomorphic to the known algebra H . These new algebras contain a nonzero omnipresent idempotent. Furthermore, we classify all four-dimensional absolute valued algebras having two dif-ferent subalgebras isomorphic to C . Note that there exists a four-dimensional absolute valued algebra containing no subalgebra of dimension two, which means that, the problem of classifying all four-dimensional absolute valued algebras seems still to be open.
{"title":"Four dimensional absolute valued algebras having two different subalgebras isomorphic to C","authors":"A. Moutassim, Mohamed Louzari, Aziz Es. Sadiq","doi":"10.12988/ija.2023.91737","DOIUrl":"https://doi.org/10.12988/ija.2023.91737","url":null,"abstract":"The aim of this paper is to construct, by algebraic methods, some new class of four-dimensional absolute valued algebras, namely M 1 , M 2 and M 3 which are not isomorphic to the known algebra H . These new algebras contain a nonzero omnipresent idempotent. Furthermore, we classify all four-dimensional absolute valued algebras having two dif-ferent subalgebras isomorphic to C . Note that there exists a four-dimensional absolute valued algebra containing no subalgebra of dimension two, which means that, the problem of classifying all four-dimensional absolute valued algebras seems still to be open.","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"6 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79800618","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}
C. Jaya, Subba Reddy, V.S.V. Krishna Murty, Preethi Budi, D. Dheeraj
In this paper we prove some results about the commutativity of the rings with involution of second kind satisfying certain conditions involving generalized reverse derivations. We also extend these results for generalized reverse derivations of prime rings to ideals. The aim of the present paper is to establish some results on commutativity of generalized reverse ∗ -derivations
{"title":"Some commutativity theorems in rings with involution involving generalized reverse derivations","authors":"C. Jaya, Subba Reddy, V.S.V. Krishna Murty, Preethi Budi, D. Dheeraj","doi":"10.12988/ija.2023.91773","DOIUrl":"https://doi.org/10.12988/ija.2023.91773","url":null,"abstract":"In this paper we prove some results about the commutativity of the rings with involution of second kind satisfying certain conditions involving generalized reverse derivations. We also extend these results for generalized reverse derivations of prime rings to ideals. The aim of the present paper is to establish some results on commutativity of generalized reverse ∗ -derivations","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"20 7 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83919417","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}
Much information about rings can be gained by studying their modules, and similarly, graded rings can be studied by studying their graded modules. And just as modules can be studied by considering their various envelopes and coverings, graded modules can be studied using the graded counterpart of these notions. Graded torsion and graded torsion free modules often arise in the study of projective geometry. Artin and Zhang [2] use graded torsion and graded torsion free modules in their discussion of noncommutative projective varieties. It is known that a module over an integral domain has a unique torsion free covering [1]. In this paper, we initiate the study of graded torsion free, graded divisible, and graded injective modules over the (graded) integral domain k[x] (where k is a field).
{"title":"On graded torsion free and graded injective k[x] modules","authors":"William Todd Ashby","doi":"10.12988/ija.2023.91735","DOIUrl":"https://doi.org/10.12988/ija.2023.91735","url":null,"abstract":"Much information about rings can be gained by studying their modules, and similarly, graded rings can be studied by studying their graded modules. And just as modules can be studied by considering their various envelopes and coverings, graded modules can be studied using the graded counterpart of these notions. Graded torsion and graded torsion free modules often arise in the study of projective geometry. Artin and Zhang [2] use graded torsion and graded torsion free modules in their discussion of noncommutative projective varieties. It is known that a module over an integral domain has a unique torsion free covering [1]. In this paper, we initiate the study of graded torsion free, graded divisible, and graded injective modules over the (graded) integral domain k[x] (where k is a field).","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"4 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82077753","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":"Relative finitistic projective and injective dimensions","authors":"Y. Wei, Xi Tang","doi":"10.12988/ija.2023.91758","DOIUrl":"https://doi.org/10.12988/ija.2023.91758","url":null,"abstract":"P.R","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"13 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84637525","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-Hernandez, Cesar R. Martinez-Garcia, Jesus A. Martinez-Nuno, M. Abel Leon-Hernandez
As a different alternative to the well established theory to solve systems of rectangular equations
{"title":"Solutions to systems of rectangular linear equations mxn, m<n","authors":"E. Salinas-Hernandez, Cesar R. Martinez-Garcia, Jesus A. Martinez-Nuno, M. Abel Leon-Hernandez","doi":"10.12988/ija.2023.91845","DOIUrl":"https://doi.org/10.12988/ija.2023.91845","url":null,"abstract":"As a different alternative to the well established theory to solve systems of rectangular equations","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135211055","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, M. A. Leon-Hernandez, Cesar R. Martinez-Garcia, J. A. Martinez-Nuno
In this work, starting from the generalized cross product
本文从广义叉乘出发
{"title":"Reduced Cramer's rule","authors":"E. Salinas-Hernández, M. A. Leon-Hernandez, Cesar R. Martinez-Garcia, J. A. Martinez-Nuno","doi":"10.12988/ija.2023.91795","DOIUrl":"https://doi.org/10.12988/ija.2023.91795","url":null,"abstract":"In this work, starting from the generalized cross product","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"27 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82315484","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}
A. Moutassim, Mohamed Mohamed Louzari, Aziz Es. Sadiq
In [9], we have proven that if A is a four-dimensional absolute valued algebra having two different subalgebras isomorphic to C , then A is isomorphic to H , M 1 , M 2 or M 3 . Here we complete the study of A . Indeed, we show if A has two different subalgebras isomorphic to ∗ C . Then A is isomorphic to ∗ H , ∗ M 1 , ∗ M 2 or ∗ M 3 . Furthermore, we classify all four-dimensional absolute valued algebras containing a nonzero idempotent commuting with all idempotents, such an algebra A contains at least two different subalgebras isomorphic to ∗ C . Which means that A is isomorphic to ∗ H , ∗ M 1 , ∗ M 2 or ∗ M 3 .
{"title":"Four dimensional absolute valued algebras containing a nonzero idempotent commuting with all idempotents","authors":"A. Moutassim, Mohamed Mohamed Louzari, Aziz Es. Sadiq","doi":"10.12988/ija.2023.91757","DOIUrl":"https://doi.org/10.12988/ija.2023.91757","url":null,"abstract":"In [9], we have proven that if A is a four-dimensional absolute valued algebra having two different subalgebras isomorphic to C , then A is isomorphic to H , M 1 , M 2 or M 3 . Here we complete the study of A . Indeed, we show if A has two different subalgebras isomorphic to ∗ C . Then A is isomorphic to ∗ H , ∗ M 1 , ∗ M 2 or ∗ M 3 . Furthermore, we classify all four-dimensional absolute valued algebras containing a nonzero idempotent commuting with all idempotents, such an algebra A contains at least two different subalgebras isomorphic to ∗ C . Which means that A is isomorphic to ∗ H , ∗ M 1 , ∗ M 2 or ∗ M 3 .","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"94 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82449752","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}
A type F of ordered semigroups is a class of ordered semigroups such that (i) if S belongs to F and S is isomorphic to S, then S belongs to F , and (ii) any one-element (ordered) semigroup belongs to F . Given a type F of ordered semigroups and an ordered semigroup S, the radS F is the intersection of all pseudoorders of S having type F (a pseudoorder σ on S has type F if the quotient semigroup of S by the congruence 1 : has also type F - we consider the quotient semigroup as an ordered semigroup under the induced order relation by σ). The derived type F of F is the class of all ordered semigroups S such that radS F is the order relation of S. An F maximal homomorphic image of an ordered semigroup of an ordered semigroup S is an ordered semigroup S in F for which there exists a homomorphism η of S onto S with the factorization property: if υ is a homomorphism of S onto an ordered semigroup of type F , then there exists a homomorphism θ of S onto T such that . We give sufficient and necessary condition under which an ordered semigroup admits an F maximal homomorphic image. We show that every ordered semigroup has an F maximal homomorphic image and finally for a type F of ordered semigroups we prove thatevery ordered semigroup has an F maximal homomorphic image if and only if
有序半群的F型是一类有序半群，满足(i)如果S属于F，且S同构于S，则S属于F， (ii)任何单元素(有序)半群属于F。给定一个有序半群的F型和一个有序半群S, radS (F)是S的所有伪阶的F型的交(S上的伪阶σ是F型，如果S的商半群具有同余1，则S上的一个伪阶σ具有F型;派生类型F (F是所有序半群类年代,拉德F S F最大同态的顺序关系有序的图像偏序半群的半群S是一个有序的半群SF存在一个同态的η到年代分解性质:如果υ年代到有序类型的半群的同态F,那么存在一个同态θ的S到T。给出了有序半群存在F极大同态象的充要条件。证明了每一个有序半群有一个F极大同态像，最后证明了对于一类F型有序半群，当且仅当每一个有序半群有一个F极大同态像
{"title":"Radicals of an ordered semigroup in terms of type of ordered semigroups","authors":"M. Tsingelis","doi":"10.12988/ija.2023.91736","DOIUrl":"https://doi.org/10.12988/ija.2023.91736","url":null,"abstract":"A type F of ordered semigroups is a class of ordered semigroups such that (i) if S belongs to F and S is isomorphic to S, then S belongs to F , and (ii) any one-element (ordered) semigroup belongs to F . Given a type F of ordered semigroups and an ordered semigroup S, the radS F is the intersection of all pseudoorders of S having type F (a pseudoorder σ on S has type F if the quotient semigroup of S by the congruence 1 : has also type F - we consider the quotient semigroup as an ordered semigroup under the induced order relation by σ). The derived type F of F is the class of all ordered semigroups S such that radS F is the order relation of S. An F maximal homomorphic image of an ordered semigroup of an ordered semigroup S is an ordered semigroup S in F for which there exists a homomorphism η of S onto S with the factorization property: if υ is a homomorphism of S onto an ordered semigroup of type F , then there exists a homomorphism θ of S onto T such that . We give sufficient and necessary condition under which an ordered semigroup admits an F maximal homomorphic image. We show that every ordered semigroup has an F maximal homomorphic image and finally for a type F of ordered semigroups we prove thatevery ordered semigroup has an F maximal homomorphic image if and only if","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73706759","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}
The classical linear eigenvalue problem is considered for matrices whose elements are dependent on a single real variable. Explicit expressions for derivatives of the eiegnvalues and eigenvectors are given in cases of simple and multiple eigenvalues. Recursion relations are obtained for derivatives of consecutively indexed generalized eigenvectors. Particular emphasis is placed on derivatives of eigenvectors and generalized eigenvectors to which enough coverage has not yet been provided in the present day literature.
{"title":"On derivatives of eigenvalues, eigenvectors and generalized eigenvectors of matrices","authors":"N. Yener","doi":"10.12988/ija.2023.91740","DOIUrl":"https://doi.org/10.12988/ija.2023.91740","url":null,"abstract":"The classical linear eigenvalue problem is considered for matrices whose elements are dependent on a single real variable. Explicit expressions for derivatives of the eiegnvalues and eigenvectors are given in cases of simple and multiple eigenvalues. Recursion relations are obtained for derivatives of consecutively indexed generalized eigenvectors. Particular emphasis is placed on derivatives of eigenvectors and generalized eigenvectors to which enough coverage has not yet been provided in the present day literature.","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"68 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84097733","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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