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New characterization of $(b,c)$-inverses through polarity 通过极性对 $(b,c)$反向的新表征
arXiv - MATH - Rings and Algebras Pub Date : 2024-09-18 DOI: arxiv-2409.11987
Btissam Laghmam, Hassane Zguitti
{"title":"New characterization of $(b,c)$-inverses through polarity","authors":"Btissam Laghmam, Hassane Zguitti","doi":"arxiv-2409.11987","DOIUrl":"https://doi.org/arxiv-2409.11987","url":null,"abstract":"In this paper we introduce the notion of $(b,c)$-polar elements in an\u0000associative ring $R$. Necessary and sufficient conditions of an element $ain\u0000R$ to be $(b,c)$-polar are investigated. We show that an element $ain R$ is\u0000$(b,c)$-polar if and only if $a$ is $(b,c)$-invertible. In particular the\u0000$(b,c)$-polarity is a generalization of the polarity along an element\u0000introduced by Song, Zhu and Mosi'c [14] if $b=c$, and the polarity introduced\u0000by Koliha and Patricio [10]. Further characterizations are obtained in the\u0000Banach space context.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142247460","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}
引用次数: 0
Relative torsionfreeness and Frobenius extensions 相对无扭和弗罗贝纽斯扩展
arXiv - MATH - Rings and Algebras Pub Date : 2024-09-18 DOI: arxiv-2409.11892
Yanhong Bao, Jiafeng Lü, Zhibing Zhao
{"title":"Relative torsionfreeness and Frobenius extensions","authors":"Yanhong Bao, Jiafeng Lü, Zhibing Zhao","doi":"arxiv-2409.11892","DOIUrl":"https://doi.org/arxiv-2409.11892","url":null,"abstract":"Let $S/R$ be a Frobenius extension with $_RS_R$ centrally projective over\u0000$R$. We show that if $_Romega$ is a Wakamatsu tilting module then so is\u0000$_SSotimes_Romega$, and the natural ring homomorphism from the endomorphism\u0000ring of $_Romega$ to the endomorphism ring of $_SSotimes_Romega$ is a\u0000Frobenius extension in addition that pd$(omega_T)$ is finite, where $T$ is the\u0000endomorphism ring of $_Romega$. We also obtain that the relative\u0000$n$-torsionfreeness of modules is preserved under Frobenius extensions.\u0000Furthermore, we give an application, which shows that the generalized\u0000G-dimension with respect to a Wakamatsu module is invariant under Frobenius\u0000extensions.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142247462","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}
引用次数: 0
Signature matrices of membranes 膜的特征矩阵
arXiv - MATH - Rings and Algebras Pub Date : 2024-09-18 DOI: arxiv-2409.11996
Felix Lotter, Leonard Schmitz
{"title":"Signature matrices of membranes","authors":"Felix Lotter, Leonard Schmitz","doi":"arxiv-2409.11996","DOIUrl":"https://doi.org/arxiv-2409.11996","url":null,"abstract":"We prove that, unlike in the case of paths, the signature matrix of a\u0000membrane does not satisfy any algebraic relations. We derive novel closed-form\u0000expressions for the signatures of polynomial membranes and piecewise bilinear\u0000interpolations for arbitrary $2$-parameter data in $d$-dimensional space. We\u0000show that these two families of membranes admit the same set of signature\u0000matrices and scrutinize the corresponding affine variety.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142247463","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}
引用次数: 0
Combinatorics of graded module categories over skew polynomial algebras at roots of unity 一元根倾斜多项式代数上分级模块类别的组合学
arXiv - MATH - Rings and Algebras Pub Date : 2024-09-17 DOI: arxiv-2409.10904
Akihiro Higashitani, Kenta Ueyama
{"title":"Combinatorics of graded module categories over skew polynomial algebras at roots of unity","authors":"Akihiro Higashitani, Kenta Ueyama","doi":"arxiv-2409.10904","DOIUrl":"https://doi.org/arxiv-2409.10904","url":null,"abstract":"We introduce an operation on skew-symmetric matrices over\u0000$mathbb{Z}/ellmathbb{Z}$ called switching, and also define a class of\u0000skew-symmetric matrices over $mathbb{Z}/ellmathbb{Z}$ referred to as modular\u0000Eulerian matrices. We then show that these are closely related to the graded\u0000module categories over skew polynomial algebras at $ell$-th roots of unity. As\u0000an application, we study the point simplicial complexes of skew polynomial\u0000algebras at cube roots of unity.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"196 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142247464","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}
引用次数: 0
On denominator conjecture for cluster algebras of finite type 关于有限类型簇代数的分母猜想
arXiv - MATH - Rings and Algebras Pub Date : 2024-09-17 DOI: arxiv-2409.10914
Changjian Fu, Shengfei Geng
{"title":"On denominator conjecture for cluster algebras of finite type","authors":"Changjian Fu, Shengfei Geng","doi":"arxiv-2409.10914","DOIUrl":"https://doi.org/arxiv-2409.10914","url":null,"abstract":"We continue our investigation on denominator conjecture of Fomin and\u0000Zelevinsky for cluster algebras via geometric models initialed in cite{FG22}.\u0000In this paper, we confirm the denominator conjecture for cluster algebras of\u0000finite type. The new contribution is a proof of this conjecture for cluster\u0000algebras of type $mathbb{D}$ and an algorithm for the exceptional types. For\u0000the type $mathbb{D}$ cases, our approach involves geometric model provided by\u0000discs with a puncture. By removing the puncture or changing the puncture to an\u0000unmarked boundary component, this also yields an alternative proof for the\u0000denominator conjecture of cluster algebras of type $mathbb{A}$ and\u0000$mathbb{C}$ respectively.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142247406","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}
引用次数: 0
Noetherianity of Diagram Algebras 图式代数的诺特性
arXiv - MATH - Rings and Algebras Pub Date : 2024-09-17 DOI: arxiv-2409.10885
Anthony Muljat, Khoa Ta
{"title":"Noetherianity of Diagram Algebras","authors":"Anthony Muljat, Khoa Ta","doi":"arxiv-2409.10885","DOIUrl":"https://doi.org/arxiv-2409.10885","url":null,"abstract":"In this short paper, we establish the local Noetherian property for the\u0000linear categories of Brauer, partition algebras, and other related categories\u0000of diagram algebras with no restrictions on their various parameters.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142247408","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}
引用次数: 0
Differential envelopes of Novikov conformal algebras 诺维科夫共形代数的微分包络
arXiv - MATH - Rings and Algebras Pub Date : 2024-09-16 DOI: arxiv-2409.10029
P. S. Kolesnikov, A. A. Nesterenko
{"title":"Differential envelopes of Novikov conformal algebras","authors":"P. S. Kolesnikov, A. A. Nesterenko","doi":"arxiv-2409.10029","DOIUrl":"https://doi.org/arxiv-2409.10029","url":null,"abstract":"A Novikov conformal algebra is a conformal algebra such that its coefficient\u0000algebra is right-symmetric and left commutative (i.e., it is an ``ordinary''\u0000Novikov algebra). We prove that every Novikov conformal algebra with a\u0000uniformly bounded locality function on a set of generators can be embedded into\u0000a commutative conformal algebra with a derivation. In particular, every\u0000finitely generated Novikov conformal algebra has a commutative conformal\u0000differential envelope. For infinitely generated algebras this statement is not\u0000true in general.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"196 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142247412","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}
引用次数: 0
On the identities and cocharacters of the algebra of $3 times 3$ matrices with orthosymplectic superinvolution 论具有正交超卷积的 3 美元乘 3 美元矩阵代数的同调与共调
arXiv - MATH - Rings and Algebras Pub Date : 2024-09-16 DOI: arxiv-2409.10187
Sara Accomando
{"title":"On the identities and cocharacters of the algebra of $3 times 3$ matrices with orthosymplectic superinvolution","authors":"Sara Accomando","doi":"arxiv-2409.10187","DOIUrl":"https://doi.org/arxiv-2409.10187","url":null,"abstract":"Let $M_{1,2}(F)$ be the algebra of $3 times 3$ matrices with orthosymplectic\u0000superinvolution $*$ over a field $F$ of characteristic zero. We study the\u0000$*$-identities of this algebra through the representation theory of the group\u0000$mathbb{H}_n = (mathbb{Z}_2 times mathbb{Z}_2) sim S_n$. We decompose the\u0000space of multilinear $*$-identities of degree $n$ into the sum of irreducibles\u0000under the $mathbb{H}_n$-action in order to study the irreducible characters\u0000appearing in this decomposition with non-zero multiplicity. Moreover, by using\u0000the representation theory of the general linear group, we determine all the\u0000$*$-polynomial identities of $M_{1,2}(F)$ up to degree $3$.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"51 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142247410","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}
引用次数: 0
Smooth geometry of double extension regular algebras of type (14641) 型双扩展正则表达式的光滑几何 (14641)
arXiv - MATH - Rings and Algebras Pub Date : 2024-09-16 DOI: arxiv-2409.10264
Andrés Rubiano, Armando Reyes
{"title":"Smooth geometry of double extension regular algebras of type (14641)","authors":"Andrés Rubiano, Armando Reyes","doi":"arxiv-2409.10264","DOIUrl":"https://doi.org/arxiv-2409.10264","url":null,"abstract":"In this paper, we prove that double extension regular algebras of type\u0000(14641) are not differentially smooth.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142247415","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}
引用次数: 0
Polynomial functions on a class of finite non-commutative rings 一类有限非交换环上的多项式函数
arXiv - MATH - Rings and Algebras Pub Date : 2024-09-16 DOI: arxiv-2409.10208
Amr Ali Abdulkader Al-Maktry, Susan F. El-Deken
{"title":"Polynomial functions on a class of finite non-commutative rings","authors":"Amr Ali Abdulkader Al-Maktry, Susan F. El-Deken","doi":"arxiv-2409.10208","DOIUrl":"https://doi.org/arxiv-2409.10208","url":null,"abstract":"Let $R$ be a finite non-commutative ring with $1ne 0$. By a polynomial\u0000function on $R$, we mean a function $Fcolon Rlongrightarrow R$ induced by a\u0000polynomial $f=sumlimits_{i=0}^{n}a_ix^iin R[x]$ via right substitution of\u0000the variable $x$, i.e. $F(a)=f(a)= sumlimits_{i=0}^{n}a_ia^i$ for every $ain R$. In this paper,\u0000we study the polynomial functions of the free $R$-algebra with a central basis\u0000${1,beta_1,ldots,beta_k}$ ($kge 1$) such that $beta_ibeta_j=0$ for\u0000every $1le i,jle k$, $R[beta_1,ldots,beta_k]$. %, the ring of dual numbers\u0000over $R$ in $k$ variables. Our investigation revolves around assigning a polynomial $lambda_f(y,z)$\u0000over $R$ in non-commutative variables $y$ and $z$ to each polynomial $f$ in\u0000$R[x]$; and describing the polynomial functions on $R[beta_1,ldots,beta_k]$\u0000through the polynomial functions induced on $R$ by polynomials in $R[x]$ and by\u0000their assigned polynomials in the in non-commutative variables $y$ and $z$.\u0000%and analyzing the resulting polynomial functions on\u0000$R[beta_1,ldots,beta_k]$. By extending results from the commutative case to the non-commutative\u0000scenario, we demonstrate that several properties and theorems in the\u0000commutative case can be generalized to the non-commutative setting with\u0000appropriate adjustments.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"75 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142247411","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}
引用次数: 0
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