Let G be a finite group. The main supergraph S(G) is a graph with vertex set G in which two vertices x and y are adjacent if and only if o(x)|o(y) or o(y)|o(x). In an earlier paper, the main properties of this graph was obtained. The aim of this paper is to investigate the Hamiltonianity, Eulerianness and 2-connectedness of this graph.
{"title":"SOME REMARKS ON THE ORDER SUPERGRAPH OF THE POWER GRAPH OF A FINITE GROUP","authors":"A. Hamzeh, A. Ashrafi","doi":"10.24330/IEJA.586838","DOIUrl":"https://doi.org/10.24330/IEJA.586838","url":null,"abstract":"Let G be a finite group. The main supergraph S(G) is a graph with vertex set G in which two vertices x and y are adjacent if and only if o(x)|o(y) or o(y)|o(x). In an earlier paper, the main properties of this graph was obtained. The aim of this paper is to investigate the Hamiltonianity, Eulerianness and 2-connectedness of this graph.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47833113","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 cotorsion theories. Among these theories we figure the pairs $(mathcal P_n,mathcal P_n^{perp})$, where $mathcal P_n$ designates the set of modules of projective dimension at most a given integer $ngeq 1$ over a ring $R$. In this paper, we shall focus on homological properties of the class $mathcal P_1^{perp}$ that we term the class of $mathcal P_1$-injective modules. Numerous nice characterizations of rings as well as of their homological dimensions arise from this study. In particular, it is shown that a ring $R$ is left hereditary if and only if any $mathcal P_1$-injective module is injective and that $R$ is left semi-hereditary if and only if any $mathcal P_1$-injective module is FP-injective. Moreover, we prove that the global dimensions of $R$ might be computed in terms of $mathcal P_1$-injective modules, namely the formula for the global dimension and the weak global dimension turn out to be as follows $$wdim(R)=sup {fd_R(M): Mmbox { is a }mathcal P_1mbox {-injective left } Rmbox {-module} }$$ and $$gdim(R)=sup {pd_R(M):M mbox { is a }mathcal P_1mbox {-injective left }Rmbox {-module}}.$$ We close the paper by proving that, given a Matlis domain $R$ and an $R$-module $Minmathcal P_1$, $Hom_R(M,N)$ is $mathcal P_1$-injective for each $mathcal P_1$-injective module $N$ if and only if $M$ is strongly flat.
有几位作者对腐蚀理论很感兴趣。在这些理论中，我们计算了对$(mathcal P_n,mathcal P_n^{perp})$，其中$mathcal P_n$表示环$R$上最多一个给定整数$ngeq 1$的射影维的模块集。在本文中，我们将重点讨论我们称之为$mathcal P_1$ -内射模块类$mathcal P_1^{perp}$的同调性质。这一研究产生了许多关于环及其同构维数的很好的特征。特别地，证明了一个环$R$是左遗传的当且仅当任何一个$mathcal P_1$ -内射模是内射，$R$是左半遗传的当且仅当任何一个$mathcal P_1$ -内射模是fp -内射。此外，我们证明了$R$的整体维数可以用$mathcal P_1$ -内射模来计算，即整体维数和弱整体维数的公式如下$$wdim(R)=sup {fd_R(M): Mmbox { is a }mathcal P_1mbox {-injective left } Rmbox {-module} }$$和$$gdim(R)=sup {pd_R(M):M mbox { is a }mathcal P_1mbox {-injective left }Rmbox {-module}}.$$。我们通过证明，给定一个Matlis域$R$和一个$R$ -模$Minmathcal P_1$，对于每个$mathcal P_1$注入模块$N$，当且仅当$M$是强平坦的时，$Hom_R(M,N)$是$mathcal P_1$注入的。
{"title":"INJECTIVE MODULES WITH RESPECT TO MODULES OF PROJECTIVE DIMENSION AT MOST ONE","authors":"S. Bouchiba, M. El-Arabi","doi":"10.24330/IEJA.586945","DOIUrl":"https://doi.org/10.24330/IEJA.586945","url":null,"abstract":"Several authors have been interested in cotorsion theories. Among these theories we figure the pairs $(mathcal P_n,mathcal P_n^{perp})$, where $mathcal P_n$ designates the set of modules of projective dimension at most a given integer $ngeq 1$ over a ring $R$. In this paper, we shall focus on homological properties of the class $mathcal P_1^{perp}$ that we term the class of $mathcal P_1$-injective modules. Numerous nice characterizations of rings as well as of their homological dimensions arise from this study. In particular, it is shown that a ring $R$ is left hereditary if and only if any $mathcal P_1$-injective module is injective and that $R$ is left semi-hereditary if and only if any $mathcal P_1$-injective module is FP-injective. Moreover, we prove that the global dimensions of $R$ might be computed in terms of $mathcal P_1$-injective modules, namely the formula for the global dimension and the weak global dimension turn out to be as follows $$wdim(R)=sup {fd_R(M): Mmbox { is a }mathcal P_1mbox {-injective left } Rmbox {-module} }$$ and $$gdim(R)=sup {pd_R(M):M mbox { is a }mathcal P_1mbox {-injective left }Rmbox {-module}}.$$ We close the paper by proving that, given a Matlis domain $R$ and an $R$-module $Minmathcal P_1$, $Hom_R(M,N)$ is $mathcal P_1$-injective for each $mathcal P_1$-injective module $N$ if and only if $M$ is strongly flat.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43289411","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 study algebraic properties of powers of squarefree principal Borel ideals I, and show that astab(I) = dstab(I). Furthermore, the behaviour of the depth function depth S/I^k is considered.
{"title":"ON THE ASSOCIATED PRIME IDEALS AND THE DEPTH OF POWERS OF SQUAREFREE PRINCIPAL BOREL IDEALS","authors":"J. Herzog, B. Lajmiri, F. Rahmati","doi":"10.24330/IEJA.587081","DOIUrl":"https://doi.org/10.24330/IEJA.587081","url":null,"abstract":"We study algebraic properties of powers of squarefree principal Borel ideals I, and show that astab(I) = dstab(I). Furthermore, the behaviour of the depth function depth S/I^k is considered.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48547085","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 article, basic ideals in a Leavitt path algebra over a com- mutative unital ring are studied. It is shown that for a nite acyclic graph E and a commutative unital ring R, the Leavitt path algebra LR(E) is a direct sum of minimal basic ideals and that for a commutative ring R and a graph E satisfying Condition (L), the Leavitt path algebra LR(E) has no non-zero nilpotent basic ideals. Uniqueness theorems for Leavitt path algebras over commutative unital rings are also discussed.
{"title":"ON LEAVITT PATH ALGEBRAS OVER COMMUTATIVE RINGS","authors":"P. Kanwar, M. Khatkar, Rajneesh Sharma","doi":"10.24330/IEJA.587053","DOIUrl":"https://doi.org/10.24330/IEJA.587053","url":null,"abstract":"In this article, basic ideals in a Leavitt path algebra over a com- mutative unital ring are studied. It is shown that for a nite acyclic graph E and a commutative unital ring R, the Leavitt path algebra LR(E) is a direct sum of minimal basic ideals and that for a commutative ring R and a graph E satisfying Condition (L), the Leavitt path algebra LR(E) has no non-zero nilpotent basic ideals. Uniqueness theorems for Leavitt path algebras over commutative unital rings are also discussed.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42714537","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 ring R is called right 2-simple J-injective if, for every 2-generated right ideal I < J(R), every R-linear map from I to R with simple image ex tends to R. The class of right 2-simple J-injective rings is broader than that of right 2-simple injective rings and right simple J-injective rings. Right 2-simple J-injective right Kasch rings are studied, several conditions under which right 2-simple J-injective rings are QF-rings are given.
环R称为右2-单J-内射，如果对于每一个2-生成的右理想I
{"title":"A GENERALIZATION OF SIMPLE-INJECTIVE RINGS","authors":"Zhu Zhanmin","doi":"10.24330/IEJA.586952","DOIUrl":"https://doi.org/10.24330/IEJA.586952","url":null,"abstract":"A ring R is called right 2-simple J-injective if, for every 2-generated right ideal I < J(R), every R-linear map from I to R with simple image ex tends to R. The class of right 2-simple J-injective rings is broader than that of right 2-simple injective rings and right simple J-injective rings. Right 2-simple J-injective right Kasch rings are studied, several conditions under which right 2-simple J-injective rings are QF-rings are given.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47842135","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}
Recognizing when a ring is a complete matrix ring is of significant importance in algebra. It is well-known folklore that a ring $R$ is a complete $ntimes n$ matrix ring, so $Rcong M_{n}(S)$ for some ring $S$, if and only if it contains a set of $ntimes n$ matrix units ${e_{ij}}_{i,j=1}^n$. A more recent and less known result states that a ring $R$ is a complete $(m+n)times(m+n)$ matrix ring if and only if, $R$ contains three elements, $a$, $b$, and $f$, satisfying the two relations $af^m+f^nb=1$ and $f^{m+n}=0$. In many instances the two elements $a$ and $b$ can be replaced by appropriate powers $a^i$ and $a^j$ of a single element $a$ respectively. In general very little is known about the structure of the ring $S$. In this article we study in depth the case $m=n=1$ when $Rcong M_2(S)$. More specifically we study the universal algebra over a commutative ring $A$ with elements $x$ and $y$ that satisfy the relations $x^iy+yx^j=1$ and $y^2=0$. We describe completely the structure of these $A$-algebras and their underlying rings when $gcd(i,j)=1$. Finally we obtain results that fully determine when there are surjections onto $M_2({mathbb F})$ when ${mathbb F}$ is a base field ${mathbb Q}$ or ${mathbb Z}_p$ for a prime number $p$.
{"title":"ON A SPECIAL PRESENTATION OF MATRIX ALGEBRAS","authors":"G. Agnarsson, S. Mendelson","doi":"10.24330/IEJA.662946","DOIUrl":"https://doi.org/10.24330/IEJA.662946","url":null,"abstract":"Recognizing when a ring is a complete matrix ring is of significant importance in algebra. It is well-known folklore that a ring $R$ is a complete $ntimes n$ matrix ring, so $Rcong M_{n}(S)$ for some ring $S$, if and only if it contains a set of $ntimes n$ matrix units ${e_{ij}}_{i,j=1}^n$. A more recent and less known result states that a ring $R$ is a complete $(m+n)times(m+n)$ matrix ring if and only if, $R$ contains three elements, $a$, $b$, and $f$, satisfying the two relations $af^m+f^nb=1$ and $f^{m+n}=0$. In many instances the two elements $a$ and $b$ can be replaced by appropriate powers $a^i$ and $a^j$ of a single element $a$ respectively. In general very little is known about the structure of the ring $S$. In this article we study in depth the case $m=n=1$ when $Rcong M_2(S)$. More specifically we study the universal algebra over a commutative ring $A$ with elements $x$ and $y$ that satisfy the relations $x^iy+yx^j=1$ and $y^2=0$. We describe completely the structure of these $A$-algebras and their underlying rings when $gcd(i,j)=1$. Finally we obtain results that fully determine when there are surjections onto $M_2({mathbb F})$ when ${mathbb F}$ is a base field ${mathbb Q}$ or ${mathbb Z}_p$ for a prime number $p$.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44373959","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 give a general method of the construction of a 3-dimensional associative algebra R over an arbitrary field F that is a sum of two subalgebras R_1 and R_2 (i.e. R = R_1 + R_2).
{"title":"ON FINITE DIMENSIONAL ALGEBRAS WHICH ARE SUMS OF TWO SUBALGEBRAS","authors":"M. Kosan, J. Žemlička","doi":"10.24330/IEJA.587018","DOIUrl":"https://doi.org/10.24330/IEJA.587018","url":null,"abstract":"In this paper, we give a general method of the construction of a 3-dimensional associative algebra R over an arbitrary field F that is a sum of two subalgebras R_1 and R_2 (i.e. R = R_1 + R_2).","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42706762","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 Noetherian ring, $I, J$ two proper ideals of $R$ and let $M$ be a non-zero finitely generated $R$-module with $c={rm cd}(I,J,M)$. In this paper, we first introduce $T_R(I,J,M)$ as the largest submodule of $M$ with the property that ${rm cd}(I,J,T_R(I,J,M))
{"title":"ANNIHILATORS OF TOP LOCAL COHOMOLOGY MODULES DEFINED BY A PAIR OF IDEALS","authors":"S. Karimi, S. Payrovi","doi":"10.24330/IEJA.586962","DOIUrl":"https://doi.org/10.24330/IEJA.586962","url":null,"abstract":"Let $R$ be a commutative Noetherian ring, $I, J$ two proper ideals of $R$ and let $M$ be a non-zero finitely generated $R$-module with $c={rm cd}(I,J,M)$. In this paper, we first introduce $T_R(I,J,M)$ as the largest submodule of $M$ with the property that ${rm cd}(I,J,T_R(I,J,M))<c$ and we describe it in terms of the reduced primary decomposition of zero submodule of $M$. It is shown that ${rm Ann}_R(H_{I,J}^d(M))={rm Ann}_R(M/{T_R(I,J,M)})$ and ${rm Ann}_R(H_{I}^d(M))={rm Ann}_R(H_{I,J}^d(M))$, whenever $R$ is a local ring, $M$ has dimension $d$ with $H_{I,J}^d(M)neq0$ and $J^tMsubseteq T_R(I,M)$ for some positive integer $t$.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46801677","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 article we introduce and study the concepts of alpha-almost quasi Artinian and alpha -quasi Krull modules. Using these concepts we extend some of the basic results of alpha -almost Artinian and alpha -Krull modules to alpha - almost quasi Artinian and alpha -quasi Krull modules. We observe that if M is an alpha -quasi Krull module then the quasi Krull dimension of M is either alpha or alpha +1.
{"title":"ON alpha-ALMOST QUASI ARTINIAN MODULES","authors":"M. Davoudian","doi":"10.24330/IEJA.586913","DOIUrl":"https://doi.org/10.24330/IEJA.586913","url":null,"abstract":"In this article we introduce and study the concepts of alpha-almost quasi Artinian and alpha -quasi Krull modules. Using these concepts we extend some of the basic results of alpha -almost Artinian and alpha -Krull modules to alpha - almost quasi Artinian and alpha -quasi Krull modules. We observe that if M is an alpha -quasi Krull module then the quasi Krull dimension of M is either alpha or alpha +1.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46949459","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 $A$ be a table algebra with standard basis $mathbf{B}$, multiplication $mu$, unit map $eta$, skew-linear involution $*$, and degree map $delta$. In this article we study the possible coalgebra structures $(A,Delta, delta)$ on $A$ for which $(A, mu, eta, Delta, delta)$ becomes a Hopf algebra with respect to some antipode. We show that such Hopf algebra structures are not always available for noncommutative table algebras. On the other hand, commutative table algebras will always have a Hopf algebra structure induced from an algebra-isomorphic group algebra. To illustrate our approach, we derive Hopf algebra comultiplications on table algebras of dimension 2 and 3.
{"title":"EXTENDING TABLE ALGEBRAS TO HOPF ALGEBRAS","authors":"A. Herman, Gurmail Singh","doi":"10.24330/IEJA.586882","DOIUrl":"https://doi.org/10.24330/IEJA.586882","url":null,"abstract":"Let $A$ be a table algebra with standard basis $mathbf{B}$, multiplication $mu$, unit map $eta$, skew-linear involution $*$, and degree map $delta$. In this article we study the possible coalgebra structures $(A,Delta, delta)$ on $A$ for which $(A, mu, eta, Delta, delta)$ becomes a Hopf algebra with respect to some antipode. We show that such Hopf algebra structures are not always available for noncommutative table algebras. On the other hand, commutative table algebras will always have a Hopf algebra structure induced from an algebra-isomorphic group algebra. To illustrate our approach, we derive Hopf algebra comultiplications on table algebras of dimension 2 and 3.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45697110","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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