A criterion for a simple object of the representation category Rep(Dω(G)) of the twisted Drinfeld double Dω(G) to be a generator is given, where G is a finite group and ω is a 3-cocycle on G. A description of the adjoint category of Rep(Dω(G)) is also given. Mathematics Subject Classification (2020): 18M20
{"title":"SOME PROPERTIES OF THE REPRESENTATION CATEGORY OF TWISTED DRINFELD DOUBLES OF FINITE GROUPS","authors":"D. Naidu","doi":"10.24330/ieja.852237","DOIUrl":"https://doi.org/10.24330/ieja.852237","url":null,"abstract":"A criterion for a simple object of the representation category Rep(Dω(G)) of the twisted Drinfeld double Dω(G) to be a generator is given, where G is a finite group and ω is a 3-cocycle on G. A description of the adjoint category of Rep(Dω(G)) is also given. Mathematics Subject Classification (2020): 18M20","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43174681","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 purpose of the present paper is to classify generalized derivations satisfying more specific algebraic identities in a prime ring with involution of the second kind. Some well-known results characterizing commutativity of prime rings by derivations have been generalized by using generalized derivation. Mathematics Subject Classification (2020): 16N60, 16W10, 16W25
{"title":"CLASSIFICATION OF SOME SPECIAL GENERALIZED DERIVATIONS","authors":"M. A. Idrissi, L. Oukhtite","doi":"10.24330/ieja.852003","DOIUrl":"https://doi.org/10.24330/ieja.852003","url":null,"abstract":"The purpose of the present paper is to classify generalized derivations satisfying more specific algebraic identities in a prime ring with involution of the second kind. Some well-known results characterizing commutativity of prime rings by derivations have been generalized by using generalized derivation. Mathematics Subject Classification (2020): 16N60, 16W10, 16W25","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47908280","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 ring, n be an non-negative integer and d be a positive integer or ∞ . A right R -module M is called ( n,d ) ∗ -projective if Ext 1 R ( M,C ) = 0 for every n -copresented right R -module C of injective dimension ≤ d ; a ring R is called right ( n,d ) -cocoherent if every n -copresented right R -module C with id ( C ) ≤ d is ( n +1)-copresented; a ring R is called right ( n,d ) -cosemihereditary if whenever 0 → C → E → A → 0 is exact, where C is n -copresented with id ( C ) ≤ d , E is finitely cogenerated injective, then A is injective; a ring R is called right ( n,d ) - V -ring if every n -copresented right R -module C with id ( C ) ≤ d is injective. Some characterizations of ( n,d ) ∗ -projective modules are given, right ( n,d )-cocoherent rings, right ( n,d )-cosemihereditary rings and right ( n,d )- V -rings are characterized by ( n,d ) ∗ -projective right R -modules. ( n,d ) ∗ -projective dimensions of modules over right ( n,d )-cocoherent rings are investigated.
{"title":"$(n,d)$-COCOHERENT RINGS, $(n,d)$-COSEMIHEREDITARY RINGS AND $(n,d)$-$V$ -RINGS","authors":"Zhu Zhanmin","doi":"10.24330/ieja.852216","DOIUrl":"https://doi.org/10.24330/ieja.852216","url":null,"abstract":". Let R be a ring, n be an non-negative integer and d be a positive integer or ∞ . A right R -module M is called ( n,d ) ∗ -projective if Ext 1 R ( M,C ) = 0 for every n -copresented right R -module C of injective dimension ≤ d ; a ring R is called right ( n,d ) -cocoherent if every n -copresented right R -module C with id ( C ) ≤ d is ( n +1)-copresented; a ring R is called right ( n,d ) -cosemihereditary if whenever 0 → C → E → A → 0 is exact, where C is n -copresented with id ( C ) ≤ d , E is finitely cogenerated injective, then A is injective; a ring R is called right ( n,d ) - V -ring if every n -copresented right R -module C with id ( C ) ≤ d is injective. Some characterizations of ( n,d ) ∗ -projective modules are given, right ( n,d )-cocoherent rings, right ( n,d )-cosemihereditary rings and right ( n,d )- V -rings are characterized by ( n,d ) ∗ -projective right R -modules. ( n,d ) ∗ -projective dimensions of modules over right ( n,d )-cocoherent rings are investigated.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44921854","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 nonzero identity and let M be a unitary R-module. The essential graph of M , denoted by EG(M) is a simple undirected graph whose vertex set is Z(M)AnnR(M) and two distinct vertices x and y are adjacent if and only if AnnM (xy) is an essential submodule of M . Let r(AnnR(M)) 6= AnnR(M). It is shown that EG(M) is a connected graph with diam(EG(M)) ≤ 2. Whenever M is Noetherian, it is shown that EG(M) is a complete graph if and only if either Z(M) = r(AnnR(M)) or EG(M) = K2 and diam(EG(M)) = 2 if and only if there are x, y ∈ Z(M)AnnR(M) and p ∈ AssR(M) such that xy 6∈ p. Moreover, it is proved that gr(EG(M)) ∈ {3,∞}. Furthermore, for a Noetherian module M with r(AnnR(M)) = AnnR(M) it is proved that |AssR(M)| = 2 if and only if EG(M) is a complete bipartite graph that is not a star. Mathematics Subject Classification (2020): 05C25, 13C99
{"title":"A GENERALIZATION OF THE ESSENTIAL GRAPH FOR MODULES OVER COMMUTATIVE RINGS","authors":"F. Soheilnia, S. Payrovi, A. Behtoei","doi":"10.24330/ieja.852234","DOIUrl":"https://doi.org/10.24330/ieja.852234","url":null,"abstract":"Let R be a commutative ring with nonzero identity and let M be a unitary R-module. The essential graph of M , denoted by EG(M) is a simple undirected graph whose vertex set is Z(M)AnnR(M) and two distinct vertices x and y are adjacent if and only if AnnM (xy) is an essential submodule of M . Let r(AnnR(M)) 6= AnnR(M). It is shown that EG(M) is a connected graph with diam(EG(M)) ≤ 2. Whenever M is Noetherian, it is shown that EG(M) is a complete graph if and only if either Z(M) = r(AnnR(M)) or EG(M) = K2 and diam(EG(M)) = 2 if and only if there are x, y ∈ Z(M)AnnR(M) and p ∈ AssR(M) such that xy 6∈ p. Moreover, it is proved that gr(EG(M)) ∈ {3,∞}. Furthermore, for a Noetherian module M with r(AnnR(M)) = AnnR(M) it is proved that |AssR(M)| = 2 if and only if EG(M) is a complete bipartite graph that is not a star. Mathematics Subject Classification (2020): 05C25, 13C99","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43374285","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 1 6= 0 and let m and n be integers with 1 ≤ n < m. A proper ideal I of R is called an (m,n)-closed ideal of R if whenever am ∈ I for some a ∈ R implies an ∈ I. Let f : A → B be a ring homomorphism and let J be an ideal of B. This paper investigates the concept of (m,n)-closed ideals in the amalgamation of A with B along J with respect f denoted by A ./f J . Namely, Section 2 investigates this notion to some extensions of ideals of A to A ./f J . Section 3 features the main result, which examines when each proper ideal of A ./f J is an (m,n)-closed ideal. This allows us to give necessary and sufficient conditions for the amalgamation to inherit the radical ideal property with applications on the transfer of von Neumann regular, π-regular and semisimple properties. Mathematics Subject Classification (2020): 13F05, 13A15, 13E05, 13F20, 13C10, 13C11, 13F30, 13D05
{"title":"ON $(m,n)$-CLOSED IDEALS IN AMALGAMATED ALGEBRA","authors":"Mohammed Issoual, N. Mahdou, M. A. S. Moutui","doi":"10.24330/ieja.852120","DOIUrl":"https://doi.org/10.24330/ieja.852120","url":null,"abstract":"Let R be a commutative ring with 1 6= 0 and let m and n be integers with 1 ≤ n < m. A proper ideal I of R is called an (m,n)-closed ideal of R if whenever am ∈ I for some a ∈ R implies an ∈ I. Let f : A → B be a ring homomorphism and let J be an ideal of B. This paper investigates the concept of (m,n)-closed ideals in the amalgamation of A with B along J with respect f denoted by A ./f J . Namely, Section 2 investigates this notion to some extensions of ideals of A to A ./f J . Section 3 features the main result, which examines when each proper ideal of A ./f J is an (m,n)-closed ideal. This allows us to give necessary and sufficient conditions for the amalgamation to inherit the radical ideal property with applications on the transfer of von Neumann regular, π-regular and semisimple properties. Mathematics Subject Classification (2020): 13F05, 13A15, 13E05, 13F20, 13C10, 13C11, 13F30, 13D05","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45303412","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}
This note aims to introduce a left adjoint functor to the functor which assigns a heap to a group. The adjunction is monadic. It is explained how one can decompose a free group functor through the previously introduced adjoint and employ it to describe a slightly different construction of free groups.
{"title":"A note on a free group. The decomposition of a free group functor through the category of heaps","authors":"Bernard Rybołowicz","doi":"10.24330/ieja.1260475","DOIUrl":"https://doi.org/10.24330/ieja.1260475","url":null,"abstract":"This note aims to introduce a left adjoint functor to the functor\u0000which assigns a heap to a group. The adjunction is monadic. It is\u0000explained how one can decompose a free group functor through the\u0000previously introduced adjoint and employ it to describe a slightly\u0000different construction of free groups.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46437165","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 introduce the first hom-associative Weyl algebras over a field of prime characteristic as a generalization of the first associative Weyl algebra in prime characteristic. First, we study properties of hom-associative algebras constructed from associative algebras by a general “twisting” procedure. Then, with the help of these results, we determine the commuter, center, nuclei, and set of derivations of the first hom-associative Weyl algebras. We also classify them up to isomorphism, and show, among other things, that all nonzero endomorphisms on them are injective, but not surjective. Last, we show that they can be described as a multi-parameter formal hom-associative deformation of the first associative Weyl algebra, and that this deformation induces a multi-parameter formal hom-Lie deformation of the corresponding Lie algebra, when using the commutator as bracket.
{"title":"The hom-associative Weyl algebras in prime characteristic","authors":"Per Back, J. Richter","doi":"10.24330/ieja.1058430","DOIUrl":"https://doi.org/10.24330/ieja.1058430","url":null,"abstract":"We introduce the first hom-associative Weyl algebras over a field of prime characteristic as a generalization of the first associative Weyl algebra in prime characteristic. First, we study properties of hom-associative algebras constructed from associative algebras by a general “twisting” procedure. Then, with the help of these results, we determine the commuter, center, nuclei, and set of derivations of the first hom-associative Weyl algebras. We also classify them up to isomorphism, and show, among other things, that all nonzero endomorphisms on them are injective, but not surjective. Last, we show that they can be described as a multi-parameter formal hom-associative deformation of the first associative Weyl algebra, and that this deformation induces a multi-parameter formal hom-Lie deformation of the corresponding Lie algebra, when using the commutator as bracket.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46827975","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}
For any commutative ring $A$ we introduce a generalization of $S$--noetherian rings using a here-ditary torsion theory $sigma$ instead of a multiplicatively closed subset $Ssubseteq{A}$. It is proved that totally noetherian w.r.t. $sigma$ is a local property, and if $A$ is a totally noetherian ring w.r.t $sigma$, then $sigma$ is of finite type.
{"title":"An extension of $S$--noetherian rings and modules","authors":"P. Jara","doi":"10.24330/ieja.1300716","DOIUrl":"https://doi.org/10.24330/ieja.1300716","url":null,"abstract":"For any commutative ring $A$ we introduce a generalization of $S$--noetherian rings using a here-ditary torsion theory $sigma$ instead of a multiplicatively closed subset $Ssubseteq{A}$. It is proved that totally noetherian w.r.t. $sigma$ is a local property, and if $A$ is a totally noetherian ring w.r.t $sigma$, then $sigma$ is of finite type.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43908311","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 identity, $S$ a multiplicatively closed subset of $R$, and $M$ be an $R$-module. In this paper, we study and investigate some properties of $S$-primary submodules of $M$. Among the other results, it is shown that this class of modules contains the family of primary (resp. $S$-prime) submodules properly.
{"title":"On S-primary submodules","authors":"H. Ansari-Toroghy, S. S. Pourmortazavi","doi":"10.24330/ieja.1058417","DOIUrl":"https://doi.org/10.24330/ieja.1058417","url":null,"abstract":"Let $R$ be a commutative ring with identity, $S$ a multiplicatively closed subset of $R$, and $M$ be an $R$-module. In this paper, we study and investigate some properties of $S$-primary submodules of $M$. Among the other results, it is shown that this class of modules contains the family of primary (resp. $S$-prime) submodules properly.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46536155","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}
Abstract of the paper: "G. Picavet and M. Picavet-L'Hermitte, Modules with finitely many submodules, Int. Electron. J. Algebra, 19 (2016), 119-131.": We characterize ring extensions $R subset S$ having FCP (FIP), where $S$ is the idealization of some $R$-module. As a by-product we exhibit characterizations of the modules that have finitely many submodules. Our tools are minimal ring morphisms, while Artinian conditions on rings are ubiquitous. $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$
论文摘要:“G. Picavet和M. Picavet- l 'Hermitte，具有有限多子模的模，Int.”电子。代数学报，19(2016)，119-131。:我们刻画了环扩展$R 子集S$具有FCP (FIP)，其中$S$是某个$R$-模的理想化。作为副产品，我们展示了具有有限多个子模块的模块的特征。我们的工具是最小环态射，而环上的阿提尼条件是普遍存在的。$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$
{"title":"AN ADDENDUM TO THE PAPER: MODULES WITH FINITELY MANY SUBMODULES","authors":"Gabriel Picavet, M. Picavet-L'Hermitte","doi":"10.24330/ieja.768272","DOIUrl":"https://doi.org/10.24330/ieja.768272","url":null,"abstract":"Abstract of the paper: \"G. Picavet and M. Picavet-L'Hermitte, Modules with finitely many submodules, Int. Electron. J. Algebra, 19 (2016), 119-131.\": We characterize ring extensions $R subset S$ having FCP (FIP), where $S$ is the idealization of some $R$-module. As a by-product we exhibit characterizations of the modules that have finitely many submodules. Our tools are minimal ring morphisms, while Artinian conditions on rings are ubiquitous. $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47101902","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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