We construct a family of non-equivalent pairwise extensions of the category of comodules of the Taft algebra, which are equivalent to representation categories of non-triangular quasi-Hopf algebras.
构造了Taft代数的模范畴的非等价对扩展族，等价于非三角拟hopf代数的表示范畴。
{"title":"Extensions of the category of comodules of the Taft algebra","authors":"Adriana MEJIA CASTANO","doi":"10.24330/ieja.1385160","DOIUrl":"https://doi.org/10.24330/ieja.1385160","url":null,"abstract":"We construct a family of non-equivalent pairwise extensions of the category of comodules of the Taft algebra, which are equivalent to representation categories of non-triangular quasi-Hopf algebras.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135155866","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 $k$ be an algebraically closed field of characteristic $2$, let $G$ be a finite group and let $B$ be the principal $2$-block of $kG$ with a dihedral or a generalised quaternion defect group $P$. Let also $calT(B)$ denote the group of splendid Morita auto-equivalences of $B$. We show that begin{align*} calT(B)cong Out_P(A)rtimes Out(P,calF), end{align*} where $Out(P,calF)$ is the group of outer automorphisms of $P$ which stabilize the fusion system $calF$ of $G$ on $P$ and $Out_P(A)$ is the group of algebra automorphisms of a source algebra $A$ of $B$ fixing $P$ modulo inner automorphisms induced by $(A^P)^times$.
{"title":"The group of splendid Morita equivalences of principal $2$-blocks with dihedral and generalised quaternion defect groups","authors":"cCisil Karaguzel, D. Yılmaz","doi":"10.24330/ieja.1402947","DOIUrl":"https://doi.org/10.24330/ieja.1402947","url":null,"abstract":"Let $k$ be an algebraically closed field of characteristic $2$, let $G$ be a finite group and let $B$ be the principal $2$-block of $kG$ with a dihedral or a generalised quaternion defect group $P$. Let also $calT(B)$ denote the group of splendid Morita auto-equivalences of $B$. We show that begin{align*} calT(B)cong Out_P(A)rtimes Out(P,calF), end{align*} where $Out(P,calF)$ is the group of outer automorphisms of $P$ which stabilize the fusion system $calF$ of $G$ on $P$ and $Out_P(A)$ is the group of algebra automorphisms of a source algebra $A$ of $B$ fixing $P$ modulo inner automorphisms induced by $(A^P)^times$.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":"6 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139370197","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 $G$ be a finite group. A subgroup $H$ is called $S$-semipermutable in $G$ if $HG_p$ = $G_pH$ for any $G_pin Syl_p(G)$ with $(|H|, p) = 1$, where $p$ is a prime number divisible $|G|$. Furthermore, $H$ is said to be $NH$-embedded in $G$ if there exists a normal subgroup $T$ of $G$ such that $HT$ is a Hall subgroup of $G$ and $H cap T leq H_{overline{s}G}$, where $H_{overline{s}G}$ is the largest $S$-semipermutable subgroup of $G$ contained in $H$, and $H$ is said to be $SS$-quasinormal in $G$ provided there is a supplement $B$ of $H$ to $G$ such that $H$ permutes with every Sylow subgroup of $B$. In this paper, we obtain some criteria for $p$-nilpotency and Supersolvability of a finite group and extend some known results concerning $NH$-embedded and $SS$-quasinormal subgroups.
{"title":"On $NH$-embedded and $SS$-quasinormal subgroups of finite groups","authors":"Weicheng ZHENG, Liang CUI, Wei MENG, Jiakuan LU","doi":"10.24330/ieja.1299719","DOIUrl":"https://doi.org/10.24330/ieja.1299719","url":null,"abstract":"Let $G$ be a finite group. A subgroup $H$ is called $S$-semipermutable in $G$ if $HG_p$ = $G_pH$ for any $G_pin Syl_p(G)$ with $(|H|, p) = 1$, where $p$ is a prime number divisible $|G|$. Furthermore, $H$ is said to be $NH$-embedded in $G$ if there exists a normal subgroup $T$ of $G$ such that $HT$ is a Hall subgroup of $G$ and $H cap T leq H_{overline{s}G}$, where $H_{overline{s}G}$ is the largest $S$-semipermutable subgroup of $G$ contained in $H$, and $H$ is said to be $SS$-quasinormal in $G$ provided there is a supplement $B$ of $H$ to $G$ such that $H$ permutes with every Sylow subgroup of $B$. In this paper, we obtain some criteria for $p$-nilpotency and Supersolvability of a finite group and extend some known results concerning $NH$-embedded and $SS$-quasinormal subgroups.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":"134 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135085558","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}
According to Dastanpour and Ghorbani, a ring $R$ is said to satisfy divisibility on ascending chains of right ideals ($A C C_{d}$) if, for every ascending chain of right ideals $I_{1} subseteq I_{2} subseteq I_{3} subseteq I_{4} subseteq ldots $ of $R$, there exists an integer $k in mathbb{N}$ such that for each $i geq k$, there exists an element $a_{i} in R$ such that $I_{i} =a_{i} I_{i +1}$. In this paper, we examine the transfer of the $A C C_{d}$-condition on ideals to trivial ring extensions. Moreover, we investigate the connection between the $A C C_{d}$ on ideals and other ascending chain conditions. For example we will prove that if $R$ is a ring with $A C C_{d}$ on ideals, then $R$ has $A C C$ on prime ideals.
{"title":"Rings with divisibility on ascending chains of ideals","authors":"Oussama Aymane Es Safi, N. Mahdou, M. Yousif","doi":"10.24330/ieja.1299720","DOIUrl":"https://doi.org/10.24330/ieja.1299720","url":null,"abstract":"According to Dastanpour and Ghorbani, a ring $R$ is said to satisfy divisibility on ascending chains of right ideals ($A C C_{d}$) if, for every ascending chain of right ideals $I_{1} subseteq I_{2} subseteq I_{3} subseteq I_{4} subseteq ldots $ of $R$, there exists an integer $k in mathbb{N}$ such that for each $i geq k$, there exists an element $a_{i} in R$ such that $I_{i} =a_{i} I_{i +1}$. In this paper, we examine the transfer of the $A C C_{d}$-condition on ideals to trivial ring extensions. Moreover, we investigate the connection between the $A C C_{d}$ on ideals and other ascending chain conditions. For example we will prove that if $R$ is a ring with $A C C_{d}$ on ideals, then $R$ has $A C C$ on prime ideals.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42582794","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 and let $M$ be an $R$-module. A proper submodule $N$ of $M$ is said to be an $r$-submodule if $amin N$ with $(0:_Ma)=0$ implies that $m in N$ for each $ain R$ and $min M$. The purpose of this paper is to introduce and investigate the dual notion of $r$-submodules of $M$.
{"title":"The dual notion of $r$-submodules of modules","authors":"Faranak Farshadifar","doi":"10.24330/ieja.1299269","DOIUrl":"https://doi.org/10.24330/ieja.1299269","url":null,"abstract":"Let $R$ be a commutative ring with identity and let $M$ be an $R$-module. \u0000A proper submodule $N$ of $M$ is said to be an $r$-submodule if \u0000$amin N$ with $(0:_Ma)=0$ implies that $m in N$ for each $ain R$ and $min M$. \u0000The purpose of this paper is to introduce and investigate the dual notion of $r$-submodules of $M$.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41918982","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 prime ring of characteristic not equal to $2$, $U$ be the Utumi quotient ring of $R$ and $C$ be the extended centroid of $R$. Let $G$ and $F$ be two generalized derivations on $R$ and $L$ be a non-central Lie ideal of $R$. If $FBig(G(u)Big)u = G(u^{2})$ for all $u in L$, then one of the following holds: (1) $G=0$. (2) There exist $p,q in U$ such that $G(x)=p x$, $F(x)=qx$ for all $x in R$ with $qp=p$. (3) $R$ satisfies $s_4$.
{"title":"Two generalized derivations on Lie ideals in prime rings","authors":"Ashutosh Pandey, B. Prajapati","doi":"10.24330/ieja.1281636","DOIUrl":"https://doi.org/10.24330/ieja.1281636","url":null,"abstract":"Let $R$ be a prime ring of characteristic not equal to $2$, $U$ be the Utumi quotient ring of $R$ and $C$ be the extended centroid of $R$. Let $G$ and $F$ be two generalized derivations on $R$ and $L$ be a non-central Lie ideal of $R$. If $FBig(G(u)Big)u = G(u^{2})$ for all $u in L$, then one of the following holds: \u0000\u0000(1) $G=0$.\u0000(2) There exist $p,q in U$ such that $G(x)=p x$, $F(x)=qx$ for all $x in R$ with $qp=p$.\u0000(3) $R$ satisfies $s_4$.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44429692","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}
An element $u$ of a ring $R$ is called textsl{unipotent} if $u-1$ is nilpotent. Two elements $a,bin R$ are called textsl{unipotent equivalent} if there exist unipotents $p,qin R$ such that $b=q^{-1}ap$. Two square matrices $A,B$ are called textsl{strongly unipotent equivalent} if there are unipotent triangular matrices $P,Q$ with $B=Q^{-1}AP$. In this paper, over commutative reduced rings, we characterize the matrices which are strongly unipotent equivalent to diagonal matrices. For $2times 2$ matrices over B'{e}zout domains, we characterize the nilpotent matrices unipotent equivalent to some multiples of $E_{12}$ and the nontrivial idempotents unipotent equivalent to $E_{11}$.
{"title":"Unipotent diagonalization of matrices","authors":"G. Călugăreanu","doi":"10.24330/ieja.1281654","DOIUrl":"https://doi.org/10.24330/ieja.1281654","url":null,"abstract":"An element $u$ of a ring $R$ is called textsl{unipotent} if $u-1$ is \u0000nilpotent. Two elements $a,bin R$ are called textsl{unipotent equivalent} \u0000if there exist unipotents $p,qin R$ such that $b=q^{-1}ap$. Two square \u0000matrices $A,B$ are called textsl{strongly unipotent equivalent} if there \u0000are unipotent triangular matrices $P,Q$ with $B=Q^{-1}AP$. \u0000In this paper, over commutative reduced rings, we characterize the matrices \u0000which are strongly unipotent equivalent to diagonal matrices. For $2times 2$ \u0000matrices over B'{e}zout domains, we characterize the nilpotent matrices \u0000unipotent equivalent to some multiples of $E_{12}$ and the nontrivial \u0000idempotents unipotent equivalent to $E_{11}$.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47166496","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 family of rings of the form frac{mathbb{Z}_{4}left langle x,y right rangle}{left langle x^2-a,y^2-b,yx-xy-2(c+dx+ey+fxy) right rangle} is investigated which contains the generalized Hamilton quaternions over $Z_4$. These rings are local rings of order 256. This family has 256 rings contained in 88 distinct isomorphism classes. Of the 88 non-isomorphic rings, 10 are minimal reversible nonsymmetric rings and 21 are minimal abelian reflexive nonsemicommutative rings. Few such examples have been identified in the literature thus far. The computational methods used to identify the isomorphism classes are also highlighted. Finally, some generalized Hamilton quaternion rings over $Z_{p^s}$ are characterized.
{"title":"Minimal rings related to generalized quaternion rings","authors":"","doi":"10.24330/ieja.1281705","DOIUrl":"https://doi.org/10.24330/ieja.1281705","url":null,"abstract":"The family of rings of the form \u0000frac{mathbb{Z}_{4}left langle x,y right rangle}{left langle x^2-a,y^2-b,yx-xy-2(c+dx+ey+fxy) right rangle} \u0000is investigated which contains the generalized Hamilton quaternions over $Z_4$. These rings are local rings of order 256. This family has 256 rings contained in 88 distinct isomorphism classes. Of the 88 non-isomorphic rings, 10 are minimal reversible nonsymmetric rings and 21 are minimal abelian reflexive nonsemicommutative rings. Few such examples have been identified in the literature thus far. The computational methods used to identify the isomorphism classes are also highlighted. Finally, some generalized Hamilton quaternion rings over $Z_{p^s}$ are characterized.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44193890","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}
Several authors have been interested in some like phantom morphisms such as $d$-phantoms, $d$-$operatorname{Ext}$-phantoms, neat-phantom morphisms, clean- cophantom morphisms, $RD$-phantom morphisms and $RD$-$operatorname{Ext}$-phantom morphisms. In this paper, we prove that these notions can be unified. We are mainly interested in proving that the majority of the existing results hold true in our general framework.
{"title":"$(n,d)$-$mathcal{X}_R$-phantom and $(n,d)$-$_Rmathcal{X}$-cophantom morphisms","authors":"Mourad Khattari, D. Bennis","doi":"10.24330/ieja.1260503","DOIUrl":"https://doi.org/10.24330/ieja.1260503","url":null,"abstract":"Several authors have been interested in some like phantom \u0000morphisms such as $d$-phantoms, \u0000$d$-$operatorname{Ext}$-phantoms, neat-phantom morphisms, clean- \u0000cophantom morphisms, $RD$-phantom morphisms and \u0000$RD$-$operatorname{Ext}$-phantom morphisms. In this paper, we \u0000prove that these notions can be unified. We are mainly interested \u0000in proving that the majority of the existing results hold true in \u0000our general framework.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42645918","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 extend the notions of friendly and solitary numbers to group theory and define friendly and solitary groups of type-1 and type-2. We provide many examples of friendly and solitary groups and study certain properties of the type-2 friends of cyclic $p$-groups, where $p$ is a prime number.
{"title":"A note on Friendly and Solitary Groups","authors":"Shubham Mittal, Gaurav Mittal, R. K. Sharma","doi":"10.24330/ieja.1260499","DOIUrl":"https://doi.org/10.24330/ieja.1260499","url":null,"abstract":"In this paper, we extend the notions of friendly and solitary \u0000numbers to group theory and define friendly and solitary groups of \u0000type-1 and type-2. We provide many examples of friendly and \u0000solitary groups and study certain properties of the type-2 friends \u0000of cyclic $p$-groups, where $p$ is a prime number.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45105687","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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