Different and distinct notions of regularity for modules exist in the literature. When these notions are restricted to commutative rings, they all coincide with the well-known von-Neumann regularity for rings. We give new characterizations of these distinct notions for modules in terms of both (weakly-)morphic modules and reduced modules. Furthermore, module theoretic settings are established where these in general distinct notions turn out to be indistinguishable.
{"title":"CHARACTERIZATIONS OF REGULAR MODULES","authors":"Philly Ivan Kimuli, D. Ssevviiri","doi":"10.24330/ieja.1224782","DOIUrl":"https://doi.org/10.24330/ieja.1224782","url":null,"abstract":"Different and distinct notions of regularity for modules exist in the literature. When these notions are restricted to commutative rings, they all coincide with the well-known von-Neumann regularity for rings. We give new characterizations of these distinct notions for modules in terms of both (weakly-)morphic modules and reduced modules. Furthermore, module theoretic settings are established where these in general distinct notions turn out to be indistinguishable.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46187533","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 classify all Rota---Baxter operators on the simple Lie conformal algebra $Cur(sl_2(mathbb{C}))$ and clarify which of them arise from the solutions to the conformal classical Yang---Baxter equation due to the connection discovered by Y. Hong and C. Bai in 2020.
{"title":"Rota---Baxter operators on $Cur(sl_2(mathbb{C}))$","authors":"V. Gubarev, R. Kozlov","doi":"10.24330/ieja.1218727","DOIUrl":"https://doi.org/10.24330/ieja.1218727","url":null,"abstract":"We classify all Rota---Baxter operators on the simple Lie conformal algebra $Cur(sl_2(mathbb{C}))$ and clarify which of them arise from the solutions to the conformal classical Yang---Baxter equation due to the connection discovered by Y. Hong and C. Bai in 2020.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47383321","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 length $ml$, index $l$ quasi-cyclic code can be viewed as a cyclic code of length $m$ over the field $mathbb F_{q^l}$ via a basis of the extension $mathbb F_{q^l}/mathbb F_{q}$. This cyclic code is an additive cyclic code. In [C. Güneri, F. Özdemir, P. Solé, On the additive cyclic structure of quasi-cyclic codes, Discrete. Math., 341 (2018), 2735-2741], authors characterize the $(l,m)$ values for one-generator quasi-cyclic codes for which it is impossible to have an $mathbb F_{q^l}$-linear image for any choice of the polynomial basis of $mathbb F_{q^l}/mathbb F_{q}$. But this characterization for some $(l,m)$ values is very intricate. In this paper, by the use of this characterization, we give a more simple characterization.
一个长度$ml$，索引$l$的准循环码可以看作是一个长度$m$的循环码，通过扩展$mathbb F_{q^l}/mathbb F_{q}$的基，在字段$mathbb F_{q}$上。这个循环码是一个加性循环码。在[C。g neri, F. Özdemir, P. sol，关于拟循环码的加性循环结构，离散。数学。对于任意选择$mathbb F_{q^l}/mathbb F_{q}$的多项式基，都不可能有$mathbb F_{q}$线性图像的一元拟循环码，[j]， 341(2018)， 2735-2741]，作者刻画了$(l,m)$值。但是对于某些$(l,m)$值，这种表征是非常复杂的。在本文中，利用这一表征，我们给出了一个更简单的表征。
{"title":"When do quasi-cyclic codes have $mathbb F_{q^l}$-linear image?","authors":"R. Nekooei, Z. Pourshafiey","doi":"10.24330/ieja.1198011","DOIUrl":"https://doi.org/10.24330/ieja.1198011","url":null,"abstract":"A length $ml$, index $l$ quasi-cyclic code can be viewed as a cyclic code of length $m$ over the field $mathbb F_{q^l}$ via a basis of the extension $mathbb F_{q^l}/mathbb F_{q}$. \u0000This cyclic code is an additive cyclic code. \u0000In [C. Güneri, F. Özdemir, P. Solé, On the additive cyclic structure of quasi-cyclic codes, Discrete. Math., 341 (2018), 2735-2741], authors characterize \u0000the $(l,m)$ values for one-generator quasi-cyclic codes for which it is \u0000impossible to have an $mathbb F_{q^l}$-linear image for any choice \u0000of the polynomial basis of $mathbb F_{q^l}/mathbb F_{q}$. \u0000But this characterization for some $(l,m)$ \u0000values is very intricate. In this paper, by the use of this characterization, we give a more simple characterization.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48919483","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 $mathbb{V}$ be a finite dimensional vector space over the field $mathbb{F}$. Let $S(mathbb{V})$ be the set of all subspaces of $mathbb{V}$ and $mathbb{A}subseteq S^*(mathbb{V})=S(mathbb{V})backslash{0}.$ In this paper, we define the Cayley subspace sum graph of $mathbb{V},$ denoted by Cay$(S^*(mathbb{V}),mathbb{A}), $ as the simple undirected graph with vertex set $S^*(mathbb{V})$ and two distinct vertices $X$ and $Y$ are adjacent if $X+Z=Y$ or $Y+Z=X$ for some $Zin mathbb{A}$. Having defined the Cayley subspace sum graph, we study about the connectedness, diameter and girth of several classes of Cayley subspace sum graphs Cay$(S^*(mathbb{V}), mathbb{A})$ for a finite dimensional vector space $mathbb{V}$ and $mathbb{A}subseteq S^*(mathbb{V})=S(mathbb{V})backslash{0}.$
{"title":"Cayley subspace sum graph of vector spaces","authors":"G. Kalaimurugan, S. Gopinath, T. Tamizh Chelvam","doi":"10.24330/ieja.1195466","DOIUrl":"https://doi.org/10.24330/ieja.1195466","url":null,"abstract":"Let $mathbb{V}$ be a finite dimensional vector space over the field $mathbb{F}$. Let $S(mathbb{V})$ be the set of all subspaces of $mathbb{V}$ and $mathbb{A}subseteq S^*(mathbb{V})=S(mathbb{V})backslash{0}.$ In this paper, we define the Cayley subspace sum graph of $mathbb{V},$ denoted by Cay$(S^*(mathbb{V}),mathbb{A}), $ as the simple undirected graph with vertex set $S^*(mathbb{V})$ and two distinct vertices $X$ and $Y$ are adjacent if $X+Z=Y$ or $Y+Z=X$ for some $Zin mathbb{A}$. Having defined the Cayley subspace sum graph, we study about the connectedness, diameter and girth of several classes of Cayley subspace sum graphs Cay$(S^*(mathbb{V}), mathbb{A})$ for a finite dimensional vector space $mathbb{V}$ and $mathbb{A}subseteq S^*(mathbb{V})=S(mathbb{V})backslash{0}.$","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49534064","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 1979, Fleury studied a class of modules with finite spanning dimension and dually a class of modules with ascending chain condition on non-small submodules was studied by Lomp and Ozcan in 2011. In the present work, we explore and investigate some new characterizations and properties of these classes of modules.
{"title":"On modules with chain condition on non-small submodules","authors":"A. K. Chaturvedi, Nirbhay Kumar","doi":"10.24330/ieja.1195509","DOIUrl":"https://doi.org/10.24330/ieja.1195509","url":null,"abstract":"In 1979, Fleury studied a class of modules with finite spanning dimension and dually a class of modules with ascending chain condition on non-small submodules was studied by Lomp and Ozcan in 2011. In the present work, we explore and investigate some new characterizations and properties of these classes of modules.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46629350","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 finite commutative ring with unity and $xin R$. We study the probability that the product of two randomly chosen elements (with replacement) of $R$ equals $x$. We denote this probability by $Prob_x (R)$. We determine some bounds for this probability and also obtain some characterizations of finite commutative rings based on this probability. Moreover, we determine the explicit computing formulas for $Prob_x (R)$ when $R=mathbb{Z}_mtimes mathbb{Z}_n$.
{"title":"On Generalized Probability in Finite Commutative Rings","authors":"S. Rehman, Muhammad Naveed Shaheryar","doi":"10.24330/ieja.1156662","DOIUrl":"https://doi.org/10.24330/ieja.1156662","url":null,"abstract":"Let $R$ be a finite commutative ring with unity and $xin R$. We study the probability that the product of two randomly chosen elements (with replacement) of $R$ equals $x$. We denote this probability by $Prob_x (R)$. We determine some bounds for this probability and also obtain some characterizations of finite commutative rings based on this probability. Moreover, we determine the explicit computing formulas for $Prob_x (R)$ when $R=mathbb{Z}_mtimes mathbb{Z}_n$.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43726942","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}
G. Kalaimurugan, P. Vignesh, M. Afkhami, Z. Barati
Let $R$ be a commutative ring without identity. The zero-divisor graph of $R,$ denoted by $Gamma(R)$ is a graph with vertex set $Z(R)setminus {0}$ which is the set of all nonzero zero-divisor elements of $R,$ and two distinct vertices $x$ and $y$ are adjacent if and only if $xy=0.$ In this paper, we characterize the rings whose zero-divisor graphs are ring graphs and outerplanar graphs. Further, we establish the planar index, ring index and outerplanar index of the zero-divisor graphs of finite commutative rings without identity.
{"title":"Planar index and outerplanar index of zero-divisor graphs of commutative rings without identity","authors":"G. Kalaimurugan, P. Vignesh, M. Afkhami, Z. Barati","doi":"10.24330/ieja.1152714","DOIUrl":"https://doi.org/10.24330/ieja.1152714","url":null,"abstract":"Let $R$ be a commutative ring without identity. The zero-divisor graph of $R,$ denoted by $Gamma(R)$ is a graph with vertex set $Z(R)setminus {0}$ which is the set of all nonzero zero-divisor elements of $R,$ and two distinct vertices $x$ and $y$ are adjacent if and only if $xy=0.$ In this paper, we characterize the rings whose zero-divisor graphs are ring graphs and outerplanar graphs. Further, we establish the planar index, ring index and outerplanar index of the zero-divisor graphs of finite commutative rings without identity.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42501494","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 work formally introduces and starts investigating the structure of matrix polynomial algebra extensions of a coefficient algebra by (elementary) matrix-variables over a ground polynomial ring in not necessary commuting variables. These matrix subalgebras of full matrix rings over polynomial rings show up in noncommutative algebraic geometry. We carefully study their (one-sided or bilateral) noetherianity, obtaining a precise lift of the Hilbert Basis Theorem when the ground ring is either a commutative polynomial ring, a free noncommutative polynomial ring or a skew polynomial ring extension by a free commutative term-ordered monoid. We equally address the natural but rather delicate question of recognising which matrix polynomial algebras are Cayley-Hamilton algebras, which are interesting noncommutative algebras arising from the study of $mathrm{Gl}_{n}$-varieties.
{"title":"The structure of matrix polynomial algebras","authors":"Bertrand Nguefack","doi":"10.24330/ieja.1151001","DOIUrl":"https://doi.org/10.24330/ieja.1151001","url":null,"abstract":"This work formally introduces and starts investigating the structure of matrix polynomial algebra extensions \u0000of a coefficient algebra by (elementary) matrix-variables over \u0000 a ground polynomial ring in not necessary commuting variables. \u0000 These matrix subalgebras of full matrix rings over polynomial rings show up \u0000 in noncommutative algebraic geometry. We carefully study their (one-sided or bilateral) noetherianity, obtaining a precise lift of the Hilbert Basis Theorem when the \u0000ground ring is either a commutative polynomial ring, a free noncommutative polynomial ring or a skew polynomial ring extension by a free commutative term-ordered monoid. \u0000We equally address the natural but rather delicate question of recognising which matrix polynomial algebras are Cayley-Hamilton algebras, \u0000which are interesting noncommutative algebras arising from the study of $mathrm{Gl}_{n}$-varieties.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48043620","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}
As an associative algebra, the Heisenberg--Weyl algebra $HWeyl$ is generated by two elements $A$, $B$ subject to the relation $AB-BA=1$. As a Lie algebra, however, where the usual commutator serves as Lie bracket, the elements $A$ and $B$ are not able to generate the whole space $HWeyl$. We identify a non-nilpotent but solvable Lie subalgebra $coreLie$ of $HWeyl$, for which, using some facts from the theory of bases for free Lie algebras, we give a presentation by generators and relations. Under this presentation, we show that, for some algebra isomorphism $isoH:HWeylintoHWeyl$, the Lie algebra $HWeyl$ is generated by the generators of $coreLie$, together with their images under $isoH$, and that $HWeyl$ is the sum of $coreLie$, $isoH(coreLie)$ and $lbrak coreLie,isoH(coreLie)rbrak$.
Heisenberg—Weyl代数$HWeyl$是由两个元素$A$， $B$根据关系$AB-BA=1$生成的。然而，作为李代数，通常的换向子作为李括号，元素$ a $和$B$不能生成整个空间$HWeyl$。利用自由李代数基论中的一些事实，利用生成子和关系式给出了一个非幂零但可解的李子代数。本文证明，对于一些代数同构$isoH:HWeyl到$ HWeyl$，李代数$HWeyl$是由$coreLie$的生成器及其$coreLie$下的图像生成的，并且$HWeyl$是$coreLie$， $isoH(coreLie)$和$lbrak coreLie，isoH(coreLie) $和$lbrak coreLie，isoH(coreLie)rbrak$的和。
{"title":"Lie structure of the Heisenberg-Weyl algebra","authors":"R. Cantuba","doi":"10.24330/ieja.1326849","DOIUrl":"https://doi.org/10.24330/ieja.1326849","url":null,"abstract":"As an associative algebra, the Heisenberg--Weyl algebra $HWeyl$ is generated by two elements $A$, $B$ subject to the relation $AB-BA=1$. As a Lie algebra, however, where the usual commutator serves as Lie bracket, the elements $A$ and $B$ are not able to generate the whole space $HWeyl$. We identify a non-nilpotent but solvable Lie subalgebra $coreLie$ of $HWeyl$, for which, using some facts from the theory of bases for free Lie algebras, we give a presentation by generators and relations. Under this presentation, we show that, for some algebra isomorphism $isoH:HWeylintoHWeyl$, the Lie algebra $HWeyl$ is generated by the generators of $coreLie$, together with their images under $isoH$, and that $HWeyl$ is the sum of $coreLie$, $isoH(coreLie)$ and $lbrak coreLie,isoH(coreLie)rbrak$.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42496817","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 present a new sufficient condition for a solubility criterion in terms of centralizers of elements. This result is a corrigendum of one of Zarrin's results. Furthermore, we extend some of K. Khoramshahi and M. Zarrin's results in the primitive case.
{"title":"On solubility of groups with finitely many centralizers","authors":"I. Lima, Caio Rodrigues","doi":"10.24330/ieja.1144159","DOIUrl":"https://doi.org/10.24330/ieja.1144159","url":null,"abstract":"In this paper we present a new sufficient condition for a solubility criterion in terms of centralizers of elements. This result is a corrigendum of one of Zarrin's results. Furthermore, we extend some of K. Khoramshahi and M. Zarrin's results in the primitive case.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43493133","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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