In this paper, we say a ring $R$ is Nil$_{ast}$-Artinian if any descending chain of nil ideals stabilizes. We first study Nil$_{ast}$-Artinian properties in terms of quotients, localizations, polynomial extensions and idealizations, and then study the transfer of Nil$_{ast}$-Artinian rings to amalgamated algebras. Besides, some examples are given to distinguish Nil$_{ast}$-Artinian rings, Nil$_{ast}$-Noetherian rings and Nil$_{ast}$-coherent rings.
{"title":"Nil$_{ast}$-Artinian rings","authors":"Xiaolei Zhang, W. Qi","doi":"10.24330/ieja.1260486","DOIUrl":"https://doi.org/10.24330/ieja.1260486","url":null,"abstract":"In this paper, we say a ring $R$ is Nil$_{ast}$-Artinian if any \u0000descending chain of nil ideals stabilizes. We first study \u0000Nil$_{ast}$-Artinian properties in terms of quotients, \u0000localizations, polynomial extensions and idealizations, and then \u0000study the transfer of Nil$_{ast}$-Artinian rings to amalgamated \u0000algebras. Besides, some examples are given to distinguish \u0000Nil$_{ast}$-Artinian rings, Nil$_{ast}$-Noetherian rings and \u0000Nil$_{ast}$-coherent rings.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41536533","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 prove that the family of binomials $x_1^{a_1} cdots x_m^{a_m}-y_1^{b_1}cdots y_n^{b_n}$ with $gcd(a_1, ldots, a_m, b_1, ldots, b_n)=1$ is irreducible by identifying the connection between the irreducibility of a binomial in ${mathbb C}[x_1, ldots, x_m, y_1, ldots, y_n]$ and ${mathbb C}(x_2, ldots, x_m, y_1, ldots, y_n)[x_1]$. Then we show that the necessary and sufficient conditions for the irreducibility of this family of binomials is equivalent to the existence of a unimodular matrix $U_i$ with integer entries such that $(a_1, ldots, a_m, b_1, ldots, b_n)^T=U_i be_i$ for $iin {1, ldots, m+n}$, where $be_i$ is the standard basis vector.
{"title":"Irreducibility of Binomials","authors":"Haohao Wang, Jerzy Wojdylo, Peter Oman","doi":"10.24330/ieja.1260484","DOIUrl":"https://doi.org/10.24330/ieja.1260484","url":null,"abstract":"In this paper, we prove that the family of binomials $x_1^{a_1} \u0000cdots x_m^{a_m}-y_1^{b_1}cdots y_n^{b_n}$ with $gcd(a_1, \u0000ldots, a_m, b_1, ldots, b_n)=1$ is irreducible by identifying \u0000the connection between the irreducibility of a binomial in \u0000${mathbb C}[x_1, ldots, x_m, y_1, ldots, y_n]$ and ${mathbb \u0000C}(x_2, ldots, x_m, y_1, ldots, y_n)[x_1]$. Then we show that \u0000the necessary and sufficient conditions for the irreducibility of \u0000this family of binomials is equivalent to the existence of a \u0000unimodular matrix $U_i$ with integer entries such that $(a_1, \u0000ldots, a_m, b_1, ldots, b_n)^T=U_i be_i$ for $iin {1, ldots, \u0000m+n}$, where $be_i$ is the standard basis vector.","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":"43991145","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 and $M$ be an $R$-module. A submodule $N$ of $M$ is called a d-submodule $($resp., an fd-submodule$)$ if $ann_R(m)subseteq ann_R(m')$ $($resp., $ann_R(F)subseteq ann_R(m'))$ for some $min N$ $($resp., finite subset $Fsubseteq N)$ and $m'in M$ implies that $m'in N.$ Many examples, characterizations, and properties of these submodules are given. Moreover, we use them to characterize modules satisfying Property T, reduced modules, and von Neumann regular modules.
{"title":"Baer submodules of modules over commutative rings","authors":"Adam Anebri, Hwankoo Kim, N. Mahdou","doi":"10.24330/ieja.1252741","DOIUrl":"https://doi.org/10.24330/ieja.1252741","url":null,"abstract":"Let $R$ be a commutative ring and $M$ be an $R$-module. A submodule $N$ of $M$ is called a d-submodule $($resp., an fd-submodule$)$ if $ann_R(m)subseteq ann_R(m')$ $($resp., $ann_R(F)subseteq ann_R(m'))$ for some $min N$ $($resp., finite subset $Fsubseteq N)$ and $m'in M$ implies that $m'in N.$ Many examples, characterizations, and properties of these submodules are given. Moreover, we use them to characterize modules satisfying Property T, reduced modules, and von Neumann regular modules.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49095827","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 provide a new and simple proof to show that a finite group in which every non-nilpotent maximal subgroup is normal is solvable.
我们提供了一个新的简单证明，证明了一个有限群，其中每个非幂零极大子群都是正规的，是可解的。
{"title":"A note on the solvability of a finite group in which every non-nilpotent maximal subgroup is normal","authors":"Wenjing Liu, Jiangtao Shi, Yunfeng Tian","doi":"10.24330/ieja.1252751","DOIUrl":"https://doi.org/10.24330/ieja.1252751","url":null,"abstract":"We provide a new and simple proof to show that a finite group in which every non-nilpotent maximal subgroup is normal is solvable.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43434475","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 the infinitesimal (in the sense of Joni and Rota) bialgebra $H_{RT}$ of planar rooted trees introduced in a previous work of two of the authors, whose coproduct is given by deletion of a vertex. We prove that its dual $H_{RT}^*$ is isomorphic to a free non unitary algebra, and give two free generating sets. Giving $H_{RT}$ a second product, we make it an infinitesimal bialgebra in the sense of Loday and Ronco, which allows to explicitly construct a projector onto its space of primitive elements, which freely generates $H_{RT}$.
{"title":"The dual of infinitesimal unitary Hopf algebras and planar rooted forests","authors":"Xiaomeng Wang, Loïc Foissy, Gao Xing","doi":"10.24330/ieja.1220707","DOIUrl":"https://doi.org/10.24330/ieja.1220707","url":null,"abstract":"We study the infinitesimal (in the sense of Joni and Rota) bialgebra $H_{RT}$ of planar rooted trees introduced in a previous work of two of the authors, whose coproduct is given by deletion of a vertex. We prove that its dual $H_{RT}^*$ is isomorphic to a free non unitary algebra, and give two free generating sets. Giving $H_{RT}$ a second product, we make it an infinitesimal bialgebra in the sense of Loday and Ronco, which allows to explicitly construct a projector onto its space of primitive elements, which freely generates $H_{RT}$.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135013228","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 article, we perform a classification of algebras with dimensions $leq$ 3 and with the property that each element is colinear with its square. The classification is complete up to properties of the ground field.
{"title":"Classification of three-dimensional isopotent algebras","authors":"Anton CEDİLNİK","doi":"10.24330/ieja.1217445","DOIUrl":"https://doi.org/10.24330/ieja.1217445","url":null,"abstract":"In the article, we perform a classification of algebras with dimensions $leq$ 3 and with the property that each element is colinear with its square. The classification is complete up to properties of the ground field.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135012941","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 an abelian group and $S$ a given multiplicatively closed subset of a commutative $G$-graded ring $A$ consisting of homogeneous elements. In this paper, we introduce and study $G$-graded $S$-Noetherian modules which are a generalization of $S$-Noetherian modules. We characterize $G$-graded $S$-Noetherian modules in terms of $S$-Noetherian modules. For instance, a $G$-graded $A$-module $M$ is $G$-graded $S$-Noetherian if and only if $M$ is $S$-Noetherian, provided $G$ is finitely generated and $S$ is countable. Also, we generalize some results on $G$-graded Noetherian rings and modules to $G$-graded $S$-Noetherian rings and modules.
设$G$是一个阿贝尔群，$S$是一个由齐次元组成的可交换$G$-阶环$ a $的给定乘闭子集。本文引入并研究了$G$分级$S$-Noetherian模，它是$S$-Noetherian模的一种推广。我们用$S$- noether模来描述$G$分级$S$- noether模。例如，$G$分级$ a $-模块$M$是$G$分级$S$-Noetherian，当且仅当$M$是$S$-Noetherian，前提是$G$是有限生成的，且$S$是可数的。同时，我们将$G$分级Noetherian环和模上的一些结果推广到$G$分级$S$-Noetherian环和模上。
{"title":"Graded S-Noetherian Modules","authors":"A. Ansari, B. K. Sharma","doi":"10.24330/ieja.1229782","DOIUrl":"https://doi.org/10.24330/ieja.1229782","url":null,"abstract":"Let $G$ be an abelian group and $S$ a given multiplicatively closed subset of a commutative $G$-graded ring $A$ consisting of homogeneous elements. In this paper, we introduce and study $G$-graded $S$-Noetherian modules which are a generalization of $S$-Noetherian modules. We characterize $G$-graded $S$-Noetherian modules in terms of $S$-Noetherian modules. For instance, a $G$-graded $A$-module $M$ is $G$-graded $S$-Noetherian if and only if $M$ is $S$-Noetherian, provided $G$ is finitely generated and $S$ is countable. Also, we generalize some results on $G$-graded Noetherian rings and modules to $G$-graded $S$-Noetherian rings and modules.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43400814","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 results concerning the structure of the ideals in the Leavitt path algebra of a (countable) directed graph with coefficients in an integral domain, such as, describing the set of generators for an ideal; the necessary and sufficient conditions for an ideal to be prime; the necessary and sufficient conditions for a Leavitt path algebra to be simple. Besides, some other interesting properties of ideal structure in a Leavitt path algebra are also mentioned.
{"title":"On some ideal structure of Leavitt path algebras with coefficients in integral domains","authors":"Trinh Thanh Deo, Vo Thanh Chi","doi":"10.24330/ieja.1229771","DOIUrl":"https://doi.org/10.24330/ieja.1229771","url":null,"abstract":"In this paper, we present results concerning the structure of the ideals in the Leavitt path algebra of a (countable) directed graph with coefficients in an integral domain, such as, describing the set of generators for an ideal; the necessary and sufficient conditions for an ideal to be prime; the necessary and sufficient conditions for a Leavitt path algebra to be simple. Besides, some other interesting properties of ideal structure in a Leavitt path algebra are also mentioned.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42932886","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 show that there is a little mistake in an implication in a paper of Bob Gilmer on rngs.
我们证明了Bob Gilmer关于环的一篇论文中的一个暗示有一个小错误。
{"title":"A little mistake in a paper by Bob Gilmer on rngs","authors":"A. Facchini, Jennifer Parolin","doi":"10.24330/ieja.1226292","DOIUrl":"https://doi.org/10.24330/ieja.1226292","url":null,"abstract":"We show that there is a little mistake in an implication in a paper of Bob Gilmer on rngs.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42226592","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}
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":" ","pages":""},"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}
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