A module M is called square-free if it contains nonon-zero is omorphic submodules A and B with A∩B= 0. Dually, Mis called dual-square-free if M has no proper submodules A and B with M=A+B and M/A∼=M/B. In this paper we show that if M=⊕i∈I Mi, then M is square-free iff each Mi is square-free and Mj and ⊕j=i∈I Mi are orthogonal. Dually, if M=⊕ni=1Mi, then M is dual-square-free iff each Mi is dual-square-free, 1⩽i⩽n, and Mj and ⊕ni=jMi are factor-orthogonal. Moreover, in the in finite case, weshow that if M=⊕i∈ISi is a direct sum of non-is omorphic simple modules, then M is a dual-square-free. In particular, if M=A⊕B where A is dual-square-free and B=⊕i∈ISi is a direct sum ofnon-isomorphic simple modules, then M is dual-square-free iff A and B are factor-orthogonal; this extends an earlier result by theauthors in [2, Proposition 2.8].
{"title":"On the direct sum of dual-square-free modules","authors":"Yasser Ibrahim, M. Yousif","doi":"10.12958/adm1807","DOIUrl":"https://doi.org/10.12958/adm1807","url":null,"abstract":"A module M is called square-free if it contains nonon-zero is omorphic submodules A and B with A∩B= 0. Dually, Mis called dual-square-free if M has no proper submodules A and B with M=A+B and M/A∼=M/B. In this paper we show that if M=⊕i∈I Mi, then M is square-free iff each Mi is square-free and Mj and ⊕j=i∈I Mi are orthogonal. Dually, if M=⊕ni=1Mi, then M is dual-square-free iff each Mi is dual-square-free, 1⩽i⩽n, and Mj and ⊕ni=jMi are factor-orthogonal. Moreover, in the in finite case, weshow that if M=⊕i∈ISi is a direct sum of non-is omorphic simple modules, then M is a dual-square-free. In particular, if M=A⊕B where A is dual-square-free and B=⊕i∈ISi is a direct sum ofnon-isomorphic simple modules, then M is dual-square-free iff A and B are factor-orthogonal; this extends an earlier result by theauthors in [2, Proposition 2.8].","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66420400","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 this paper, we study on the capability of groups of order p3q, where pandqare distinct prime numbers and p>2.
本文研究了p3q阶群的能力，其中p3q阶群是不同素数和p3q阶群。
{"title":"Capable groups of order p3q","authors":"O. Kalteh, S. Jafari","doi":"10.12958/adm1659","DOIUrl":"https://doi.org/10.12958/adm1659","url":null,"abstract":"In this paper, we study on the capability of groups of order p3q, where pandqare distinct prime numbers and p>2.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66418563","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 it is shown that a prime ring R with center Z satisfies s4, the standard identity in fourvariables if R admits a non-identity automorphismσsuch that [u, v]−um[uσ,u]nuσ∈Z for all u in some noncentral ideal L of R, whenever char (R)>n+m or char(R)=0, where n and m are fixed positive integer.
{"title":"An identity on automorphisms of Lie ideals in prime rings","authors":"N. Rehmam","doi":"10.12958/adm1612","DOIUrl":"https://doi.org/10.12958/adm1612","url":null,"abstract":"In the present paper it is shown that a prime ring R with center Z satisfies s4, the standard identity in fourvariables if R admits a non-identity automorphismσsuch that [u, v]−um[uσ,u]nuσ∈Z for all u in some noncentral ideal L of R, whenever char (R)>n+m or char(R)=0, where n and m are fixed positive integer.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66418684","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 In be the set of partial one-to-one transformations on the chain Xn={1,2, . . . , n} and, for each α in In, let h(α)=|Imα|, f(α)=|{x∈Xn:xα=x}| and w(α)=max(Imα). In this note, we obtain formulae involving binomial coefficients of F(n; p, m, k)=|{α ∈ In:h(α)=p∧f(α)=m∧w(α)=k}| and F(n;·, m, k)=|{α ∈ In:f(α)=m∧w(α)=k}| and analogous results on the set of partial derangements of In.
{"title":"Further combinatorial results for the symmetric inverse monoid","authors":"A. Laradji, A. Umar","doi":"10.12958/adm1793","DOIUrl":"https://doi.org/10.12958/adm1793","url":null,"abstract":"Let In be the set of partial one-to-one transformations on the chain Xn={1,2, . . . , n} and, for each α in In, let h(α)=|Imα|, f(α)=|{x∈Xn:xα=x}| and w(α)=max(Imα). In this note, we obtain formulae involving binomial coefficients of F(n; p, m, k)=|{α ∈ In:h(α)=p∧f(α)=m∧w(α)=k}| and F(n;·, m, k)=|{α ∈ In:f(α)=m∧w(α)=k}| and analogous results on the set of partial derangements of In.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"14 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66420329","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 doppelsemigroup (G,⊣,⊢) is calledcyclic if (G,⊣) is a cyclic group. In the paper, we describe up to isomorphism all cyclic (strong) doppelsemigroups. We prove that up to isomorphism there exist τ(n) finite cyclic (strong) doppelsemigroups of order n, where τ is the number of divisors function. Also there exist infinite countably many pairwise non-isomorphic infinite cyclic (strong) doppelsemigroups.
{"title":"Note on cyclic doppelsemigroups","authors":"V. Gavrylkiv","doi":"10.12958/adm1991","DOIUrl":"https://doi.org/10.12958/adm1991","url":null,"abstract":"A doppelsemigroup (G,⊣,⊢) is calledcyclic if (G,⊣) is a cyclic group. In the paper, we describe up to isomorphism all cyclic (strong) doppelsemigroups. We prove that up to isomorphism there exist τ(n) finite cyclic (strong) doppelsemigroups of order n, where τ is the number of divisors function. Also there exist infinite countably many pairwise non-isomorphic infinite cyclic (strong) doppelsemigroups.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66420605","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 this paper it is proved that the algorithms of differentiation VIII-X (introduced by A.G. Zavadskij to classify equipped posets of tame representation type) induce categorical equivalences between some quotient categories, in particular, analgorithm Az is introduced to build equipped posets with a pair ofpoints (a, b) suitable for differentiation VII such that the subset of strong points related with the weak pointais not empty.
{"title":"Categorical properties of some algorithms of differentiation for equipped posets","authors":"Isaías David Marín Gaviria, A. M. Cañadas","doi":"10.12958/adm1647","DOIUrl":"https://doi.org/10.12958/adm1647","url":null,"abstract":"In this paper it is proved that the algorithms of differentiation VIII-X (introduced by A.G. Zavadskij to classify equipped posets of tame representation type) induce categorical equivalences between some quotient categories, in particular, analgorithm Az is introduced to build equipped posets with a pair ofpoints (a, b) suitable for differentiation VII such that the subset of strong points related with the weak pointais not empty.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66418948","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 main purpose of this article is to classify all indecomposable quasi semiprime multiplication modules over pullback rings of two Dedekind domains and establish a connection between the quasi semiprime multiplication modules and the pure-injective modules over such rings. First, we introduce and study the notion of quasi semiprime multiplication modules and classify quasi semiprime multiplication modules over local Dedekind domains. Second, we get all indecomposable separated quasi semiprime multiplication modules and then, using this list of separated quasi-semiprime multiplication modules, we show that non-separated indecomposable quasi semiprime multiplication R-modules with finite-dimensional top are factor modules of finite direct sums of separated indecomposable quasi semiprime multiplication modules.
{"title":"Quasi semiprime multiplication modules over a pullback of a pair of Dedekind domains","authors":"P. Ghiasvand, F. Farzalipour","doi":"10.12958/adm1762","DOIUrl":"https://doi.org/10.12958/adm1762","url":null,"abstract":"The main purpose of this article is to classify all indecomposable quasi semiprime multiplication modules over pullback rings of two Dedekind domains and establish a connection between the quasi semiprime multiplication modules and the pure-injective modules over such rings. First, we introduce and study the notion of quasi semiprime multiplication modules and classify quasi semiprime multiplication modules over local Dedekind domains. Second, we get all indecomposable separated quasi semiprime multiplication modules and then, using this list of separated quasi-semiprime multiplication modules, we show that non-separated indecomposable quasi semiprime multiplication R-modules with finite-dimensional top are factor modules of finite direct sums of separated indecomposable quasi semiprime multiplication modules.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66419990","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 σ-parastrophe of a class of quasigroups A is a class σA of all σ-parastrophes of quasigroups from A. A set of all pairwise parastrophic classes is called a parastrophic orbit or a truss. A parastrophically closed semi-lattice of classes is a bunch. A linearity bunch is a set of varieties which contains the variety of all left linear quasigroups, the variety of all left alinear quasigroups, all their parastrophes and all their intersections. It contains 14 varieties, which are distributed into six parastrophic orbits. All quasigroups from these varieties are called dilinear. To obtain all varieties from the bunch, concepts of middle linearity and middle alinearity are introduced. A well-known identity or a system of identities which describes a variety from every parastrophic orbit of the bunch is cited. An algorithm for obtaining identities which describe all varieties from the parastrophic orbits is given. Examples of quasigroups distinguishing one variety from the other are presented.
{"title":"Semi-lattice of varieties of quasigroups with linearity","authors":"F. Sokhatsky, H. Krainichuk, V. Sydoruk","doi":"10.12958/adm1748","DOIUrl":"https://doi.org/10.12958/adm1748","url":null,"abstract":"A σ-parastrophe of a class of quasigroups A is a class σA of all σ-parastrophes of quasigroups from A. A set of all pairwise parastrophic classes is called a parastrophic orbit or a truss. A parastrophically closed semi-lattice of classes is a bunch. A linearity bunch is a set of varieties which contains the variety of all left linear quasigroups, the variety of all left alinear quasigroups, all their parastrophes and all their intersections. It contains 14 varieties, which are distributed into six parastrophic orbits. All quasigroups from these varieties are called dilinear. To obtain all varieties from the bunch, concepts of middle linearity and middle alinearity are introduced. A well-known identity or a system of identities which describes a variety from every parastrophic orbit of the bunch is cited. An algorithm for obtaining identities which describe all varieties from the parastrophic orbits is given. Examples of quasigroups distinguishing one variety from the other are presented.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46905507","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 with multiplicative identity and P is a finitely generated projective R-module. If P∗ is the set of R-module homomorphism from P to R, then the tensor product P∗⊗RP can be considered as an R-coalgebra. Furthermore, P and P∗ is a comodule over coalgebra P∗⊗RP. Using the Morita context, this paper give sufficient conditions of clean coalgebra P∗⊗RP and clean P∗⊗RP-comodule P and P∗. These sufficient conditions are determined by the conditions of module P and ring R.
{"title":"Clean coalgebras and clean comodules of finitely generated projective modules","authors":"N. P. Puspita, I. E. Wijayanti, B. Surodjo","doi":"10.12958/ADM1415","DOIUrl":"https://doi.org/10.12958/ADM1415","url":null,"abstract":"Let R be a commutative ring with multiplicative identity and P is a finitely generated projective R-module. If P∗ is the set of R-module homomorphism from P to R, then the tensor product P∗⊗RP can be considered as an R-coalgebra. Furthermore, P and P∗ is a comodule over coalgebra P∗⊗RP. Using the Morita context, this paper give sufficient conditions of clean coalgebra P∗⊗RP and clean P∗⊗RP-comodule P and P∗. These sufficient conditions are determined by the conditions of module P and ring R.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43127262","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 prove a particular case of the conjecture of Berest--Eshmatov--Eshmatov by showing that the group of unimodular automorphisms of C[x,y] acts in an infinitely-transitive way on the Calogero-Moser space C2.
{"title":"Infinite transitivity on the Calogero-Moser space C2","authors":"J. Kesten, S. Mathers, Z. Normatov","doi":"10.12958/ADM1656","DOIUrl":"https://doi.org/10.12958/ADM1656","url":null,"abstract":"We prove a particular case of the conjecture of Berest--Eshmatov--Eshmatov by showing that the group of unimodular automorphisms of C[x,y] acts in an infinitely-transitive way on the Calogero-Moser space C2.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45742312","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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