In this paper, some characterizations of partial isometries, normal elements and strongly $EP$ elements are given by the construction of $EP$ elements. At the same time, the partial isometry elements are characterized by the existence of solutions of equations in rings in a given set, and also by the general form of solutions of given equations.
{"title":"SOME NEW CHARACTERIZATIONS OF PARTIAL ISOMETRIES IN RINGS WITH INVOLUTION","authors":"Dandan Zhao, Junchao Wei","doi":"10.24330/IEJA.969942","DOIUrl":"https://doi.org/10.24330/IEJA.969942","url":null,"abstract":"In this paper, some characterizations of partial isometries, normal elements and strongly $EP$ elements are given by the construction of $EP$ elements. At the same time, the partial isometry elements are characterized by the existence of solutions of equations in rings in a given set, and also by the general form of solutions of given equations.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46517933","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 note, we show that any epimorphism originating at a von Neumann regular ring (not necessary commutative) is a universal localization. As an application, we prove that the Telescope Conjecture holds for the unbounded derived categories of von Neumann regular rings (not necessary commutative).
{"title":"The Telescope Conjecture for von Neumann regular rings","authors":"Xiaolei Zhang","doi":"10.24330/ieja.1298175","DOIUrl":"https://doi.org/10.24330/ieja.1298175","url":null,"abstract":"In this note, we show that any epimorphism originating at a von Neumann regular ring (not necessary commutative) is a universal localization. As an application, we prove that the Telescope Conjecture holds for the unbounded derived categories of von Neumann regular rings (not necessary commutative).","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46656105","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 $C_n$, $Q_n$ and $D_n$ be the cyclic group, the quaternion group and the dihedral group of order $n$, respectively. Recently, the structures of the unit groups of the finite group algebras of $2$-groups that contain a normal cyclic subgroup of index $2$ have been studied. The dihedral groups $D_{2n}, ngeq 3$ and the generalized quaternion groups $Q_{4n}, ngeq 2$ also contain a normal cyclic subgroup of index $2$. The structures of the unit groups of the finite group algebras $FQ_{12}$, $FD_{12}$, $F(C_2 times Q_{12})$ and $F(C_2 times D_{12})$ over a finite field $F$ are well known. In this article, we continue this investigation and establish the structures of the unit groups of the group algebras $F(C_n times Q_{12})$ and $F(C_n times D_{12})$ over a finite field $F$ of characteristic $p$ containing $p^k$ elements.
{"title":"Units in $F(C_n times Q_{12})$ and $F(C_n times D_{12})$","authors":"M. Sahai, S. F. Ansari","doi":"10.24330/ieja.1299278","DOIUrl":"https://doi.org/10.24330/ieja.1299278","url":null,"abstract":"Let $C_n$, $Q_n$ and $D_n$ be the cyclic group, the quaternion group and the dihedral\u0000group of order $n$, respectively. Recently, the structures of the unit groups of the finite group algebras of $2$-groups that contain a normal cyclic subgroup of index $2$ have been studied. The dihedral groups $D_{2n}, ngeq 3$ and the generalized quaternion groups $Q_{4n}, ngeq 2$ also contain a normal cyclic subgroup of index $2$. The structures of the unit groups of the finite group algebras $FQ_{12}$, $FD_{12}$, $F(C_2 times Q_{12})$ and $F(C_2 times D_{12})$ over a finite field $F$ are well known. In this article, we continue this investigation and establish the structures of the unit groups of the group algebras $F(C_n times Q_{12})$ and $F(C_n times D_{12})$ over a finite field $F$ of characteristic $p$ containing $p^k$ elements.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44537845","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 a field and $S = K[x_1,dots,x_n]$ be a polynomial ring over $K$. We discuss the behaviour of the extremal Betti numbers of the class of squarefree strongly stable ideals. More precisely, we give a numerical characterization of the possible extremal Betti numbers (values as well as positions) of such a class of squarefree monomial ideals.
{"title":"ON THE EXTREMAL BETTI NUMBERS OF SQUAREFREE MONOMIAL IDEALS","authors":"Luca Amata, M. Crupi","doi":"10.24330/ieja.969656","DOIUrl":"https://doi.org/10.24330/ieja.969656","url":null,"abstract":"Let $K$ be a field and $S = K[x_1,dots,x_n]$ be a polynomial ring over $K$. We discuss the behaviour of the extremal Betti numbers of the class of squarefree strongly stable ideals. More precisely, we give a numerical characterization of the possible extremal Betti numbers (values as well as positions) of such a class of squarefree monomial ideals.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45138760","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 first aim of this work is to characterize when the lattice of all submodules of a module is a direct product of two lattices. In particular, which decompositions of a module $M$ produce these decompositions: the emph{lattice decompositions}. In a first textit{etage} this can be done using endomorphisms of $M$, which produce a decomposition of the ring $textrm{End}_R(M)$ as a product of rings, i.e., they are central idempotent endomorphisms. But since not every central idempotent endomorphism produces a lattice decomposition, the classical theory is not of application. In a second step we characterize when a particular module $M$ has a lattice decomposition; this can be done, in the commutative case in a simple way using the support, $textrm{Supp}(M)$, of $M$; but, in general, it is not so easy. Once we know when a module decomposes, we look for characterizing its decompositions. We show that a good framework for this study, and its generalizations, could be provided by the category $sigma[M]$, the smallest Grothendieck subcategory of $textbf{Mod}-{R}$ containing $M$.
{"title":"LATTICE DECOMPOSITION OF MODULES","authors":"J. M. Garc'ia, P. Jara, L. Merino","doi":"10.24330/IEJA.969940","DOIUrl":"https://doi.org/10.24330/IEJA.969940","url":null,"abstract":"The first aim of this work is to characterize when the lattice of all submodules of a module is a direct product of two lattices. In particular, which decompositions of a module $M$ produce these decompositions: the emph{lattice decompositions}. In a first textit{etage} this can be done using endomorphisms of $M$, which produce a decomposition of the ring $textrm{End}_R(M)$ as a product of rings, i.e., they are central idempotent endomorphisms. But since not every central idempotent endomorphism produces a lattice decomposition, the classical theory is not of application. In a second step we characterize when a particular module $M$ has a lattice decomposition; this can be done, in the commutative case in a simple way using the support, $textrm{Supp}(M)$, of $M$; but, in general, it is not so easy. Once we know when a module decomposes, we look for characterizing its decompositions. We show that a good framework for this study, and its generalizations, could be provided by the category $sigma[M]$, the smallest Grothendieck subcategory of $textbf{Mod}-{R}$ containing $M$.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44013401","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 introduce a concept of X-element with respect to an M -closed set X in multiplicative lattices and study properties of X-elements. For a particular M -closed subset X, we define the concept of r-element, n-element and J-element. These elements generalize the notion of r-ideals, n-ideals and J-ideals of a commutative ring with unity to multiplicative lattices. In fact, we prove that an ideal I of a commutative ring R with unity is a n-ideal (J-ideal) of R if and only if it is an n-element (J-element) of Id(R), the ideal lattice of R.
{"title":"$mathfrak{X}$-elements in multiplicative lattices - A generalization of $J$-ideals, $n$-ideals and $r$-ideals in rings","authors":"Sachin Sarode, Vinayak Joshi","doi":"10.24330/ieja.1102289","DOIUrl":"https://doi.org/10.24330/ieja.1102289","url":null,"abstract":"In this paper, we introduce a concept of X-element with respect to an M -closed set X in multiplicative lattices and study properties of X-elements. For a particular M -closed subset X, we define the concept of r-element, n-element and J-element. These elements generalize the notion of r-ideals, n-ideals and J-ideals of a commutative ring with unity to multiplicative lattices. In fact, we prove that an ideal I of a commutative ring R with unity is a n-ideal (J-ideal) of R if and only if it is an n-element (J-element) of Id(R), the ideal lattice of R.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42906927","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 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":" ","pages":""},"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}
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