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Nil modules and the envelope of a submodule 无模块和子模块包络
arXiv - MATH - Rings and Algebras Pub Date : 2024-08-29 DOI: arxiv-2408.16240
David Ssevviiri, Annet Kyomuhangi
{"title":"Nil modules and the envelope of a submodule","authors":"David Ssevviiri, Annet Kyomuhangi","doi":"arxiv-2408.16240","DOIUrl":"https://doi.org/arxiv-2408.16240","url":null,"abstract":"Let $R$ be a commutative unital ring and $N$ be a submodule of an $R$-module\u0000$M$. The submodule $langle E_M(N)rangle$ generated by the envelope $E_M(N)$\u0000of $N$ is instrumental in studying rings and modules that satisfy the radical\u0000formula. We show that: 1) the semiprime radical is an invariant on all the\u0000submodules which are respectively generated by envelopes in the ascending chain\u0000of envelopes of a given submodule; 2) for rings that satisfy the radical\u0000formula, $langle E_M(0)rangle$ is an idempotent radical and it induces a\u0000torsion theory whose torsion class consists of all nil $R$-modules and the\u0000torsionfree class consists of all reduced $R$-modules; 3) Noetherian uniserial\u0000modules satisfy the semiprime radical formula and their semiprime radical is a\u0000nil module; and lastly, 4) we construct a sheaf of nil $R$-modules on\u0000$text{Spec}(R)$.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192730","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
PBW-deformations of smash products involving Hopf algebra of Kac-Paljutkin type 涉及 Kac-Paljutkin 型霍普夫代数的粉碎积的 PBW 变形
arXiv - MATH - Rings and Algebras Pub Date : 2024-08-29 DOI: arxiv-2408.16557
Yujie Gao, Shilin Yang
{"title":"PBW-deformations of smash products involving Hopf algebra of Kac-Paljutkin type","authors":"Yujie Gao, Shilin Yang","doi":"arxiv-2408.16557","DOIUrl":"https://doi.org/arxiv-2408.16557","url":null,"abstract":"In this paper, we present the PBW-deformations of the smash product $A #\u0000H_{2n^2}$ and the braided tensor product $(A otimes^c A^{mathrm{op}}_c) #\u0000H_{2n^2}$ for the Kac-Paljutkin type Hopf algebra $H_{2n^2}$ and its Koszul\u0000$H_{2n^2}$-module algebra $A$. A necessary and sufficient condition for the\u0000Koszul dual algebra $A^{!}$ to be an $H$-module algebra is given. Then we\u0000provide all the PBW-deformations of the smash product algebra $A^! # H_{2n^2}$\u0000for a given $H_{2n^2}$-module algebra $A$.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"68 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192727","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 Modular Invariants of Truncated Polynomial Rings in low ranks 论低级截断多项式环的模块不变式
arXiv - MATH - Rings and Algebras Pub Date : 2024-08-29 DOI: arxiv-2408.16250
Le Minh Ha, Nguyen Dang Ho Hai, Nguyen Van Nghia
{"title":"On Modular Invariants of Truncated Polynomial Rings in low ranks","authors":"Le Minh Ha, Nguyen Dang Ho Hai, Nguyen Van Nghia","doi":"arxiv-2408.16250","DOIUrl":"https://doi.org/arxiv-2408.16250","url":null,"abstract":"We verify the conjectures due to Lewis, Reiner and Stanton about the Hilbert\u0000series of the invariant ring of the truncated polynomial ring for all parabolic\u0000subgroups up to rank $3$. This is done by constructing an explicit set of\u0000generators for each invariant ring in question. We also propose a conjecture\u0000concerning the action of the Steenrod algebra and the Dickson algebra on a\u0000certain naturally occurring filtration of the invariant ring under the general\u0000linear group.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"2012 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192729","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
Direct finiteness of representable regular rings with involution: A counterexample 带内卷的可表示正则环的直接有限性：一个反例
arXiv - MATH - Rings and Algebras Pub Date : 2024-08-29 DOI: arxiv-2408.16437
Christian Herrmann
{"title":"Direct finiteness of representable regular rings with involution: A counterexample","authors":"Christian Herrmann","doi":"arxiv-2408.16437","DOIUrl":"https://doi.org/arxiv-2408.16437","url":null,"abstract":"Bruns and Roddy constructed a $3$-generated modular ortholattice $L$ which\u0000cannot be embedded into any complete modular ortholattice. Motivated by their\u0000approach, we use shift operators to construct a $*$-regular $*$-ring $R$ of\u0000endomorphisms of an inner product space (which can be chosen as the Hilbert\u0000space $ell^2$) such that direct finiteness fails for $R$.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"53 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192728","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
Rational equivalence on adjoint groups of type $^{1}D_n$ over field $mathbb{Q}_P(X)$ 域$^{1}D_n$上$^{1}D_n$型邻接群的有理等价性
arXiv - MATH - Rings and Algebras Pub Date : 2024-08-28 DOI: arxiv-2408.15528
M. Archita
{"title":"Rational equivalence on adjoint groups of type $^{1}D_n$ over field $mathbb{Q}_P(X)$","authors":"M. Archita","doi":"arxiv-2408.15528","DOIUrl":"https://doi.org/arxiv-2408.15528","url":null,"abstract":"Let $F$ be the function field of a smooth, geometrically integral curve over\u0000a $p$-adic field with $pneq 2.$ Let $G$ be a classical adjoint group of type\u0000$^1D_n$ defined over $F$. We show that $G(F) / R$ is trivial, where $R$ denotes {it rational\u0000equivalence} on $G(F)$.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192732","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
An upper bound for polynomial volume growth of automorphisms of zero entropy 零熵自形体的多项式体积增长上限
arXiv - MATH - Rings and Algebras Pub Date : 2024-08-28 DOI: arxiv-2408.15804
Fei Hu, Chen Jiang
{"title":"An upper bound for polynomial volume growth of automorphisms of zero entropy","authors":"Fei Hu, Chen Jiang","doi":"arxiv-2408.15804","DOIUrl":"https://doi.org/arxiv-2408.15804","url":null,"abstract":"Let $X$ be a normal projective variety of dimension $d$ over an algebraically\u0000closed field and $f$ an automorphism of $X$. Suppose that the pullback\u0000$f^*|_{mathsf{N}^1(X)_mathbf{R}}$ of $f$ on the real N'eron--Severi space\u0000$mathsf{N}^1(X)_mathbf{R}$ is unipotent and denote the index of the\u0000eigenvalue $1$ by $k+1$. We prove an upper bound for the polynomial volume\u0000growth $mathrm{plov}(f)$ of $f$ as follows: [ mathrm{plov}(f) le (k/2 +\u00001)d. ] This inequality is optimal in certain cases. Furthermore, we show that\u0000$kle 2(d-1)$, extending a result of Dinh--Lin--Oguiso--Zhang for compact\u0000K\"ahler manifolds to arbitrary characteristic. Combining these two\u0000inequalities together, we obtain an optimal inequality that [ mathrm{plov}(f)\u0000le d^2, ] which affirmatively answers questions of Cantat--Paris-Romaskevich\u0000and Lin--Oguiso--Zhang.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"75 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192733","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
Formalizing Mason-Stothers Theorem and its Corollaries in Lean 4 在精益 4 中实现马森-斯托瑟定理及其推论的形式化
arXiv - MATH - Rings and Algebras Pub Date : 2024-08-27 DOI: arxiv-2408.15180
Jineon Baek, Seewoo Lee
{"title":"Formalizing Mason-Stothers Theorem and its Corollaries in Lean 4","authors":"Jineon Baek, Seewoo Lee","doi":"arxiv-2408.15180","DOIUrl":"https://doi.org/arxiv-2408.15180","url":null,"abstract":"The ABC conjecture implies many conjectures and theorems in number theory,\u0000including the celebrated Fermat's Last Theorem. Mason-Stothers Theorem is a\u0000function field analogue of the ABC conjecture that admits a much more\u0000elementary proof with many interesting consequences, including a polynomial\u0000version of Fermat's Last Theorem. While years of dedicated effort are expected\u0000for a full formalization of Fermat's Last Theorem, the simple proof of\u0000Mason-Stothers Theorem and its corollaries calls for an immediate\u0000formalization. We formalize an elementary proof of by Snyder in Lean 4, and also formalize\u0000many consequences of Mason-Stothers, including (i) non-solvability of\u0000Fermat-Cartan equations in polynomials, (ii) non-parametrizability of a certain\u0000elliptic curve, and (iii) Davenport's Theorem. We compare our work to existing\u0000formalizations of Mason-Stothers by Eberl in Isabelle and Wagemaker in Lean 3\u0000respectively. Our formalization is based on the mathlib4 library of Lean 4, and\u0000is currently being ported back to mathlib4.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192734","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
Transposed Poisson structures on the extended Schrödinger-Virasoro and the original deformative Schrödinger-Virasoro algebras 扩展薛定谔-维拉索罗代数和原始变形薛定谔-维拉索罗代数上的变换泊松结构
arXiv - MATH - Rings and Algebras Pub Date : 2024-08-26 DOI: arxiv-2408.14160
Zarina Shermatova
{"title":"Transposed Poisson structures on the extended Schrödinger-Virasoro and the original deformative Schrödinger-Virasoro algebras","authors":"Zarina Shermatova","doi":"arxiv-2408.14160","DOIUrl":"https://doi.org/arxiv-2408.14160","url":null,"abstract":"We compute 1/2-derivations on the extended Schr\"odinger-Virasoro algebras\u0000and the original deformative Schr\"odinger-Virasoro algebras. The extended\u0000Schr\"odinger-Virasoro algebras have neither nontrivial 1/2-derivations nor\u0000nontrivial transposed Poisson algebra structures. We demonstrate that the\u0000original deformative Schr\"odinger-Virasoro algebras have nontrivial\u00001/2-derivations, indicating that they possess nontrivial transposed Poisson\u0000structures.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"90 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192736","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
The Krull-Remak-Schmidt decomposition of commutative group algebras 交换群代数的 Krull-Remak-Schmidt 分解
arXiv - MATH - Rings and Algebras Pub Date : 2024-08-26 DOI: arxiv-2408.14665
Robert Christian Subroto
{"title":"The Krull-Remak-Schmidt decomposition of commutative group algebras","authors":"Robert Christian Subroto","doi":"arxiv-2408.14665","DOIUrl":"https://doi.org/arxiv-2408.14665","url":null,"abstract":"We provide the Krull-Remak-Schmidt decomposition of group algebras of the\u0000form $k[G]$ where $k$ is a field, which includes fields with prime\u0000characteristic, and $G$ a finite abelian group. We achieved this by studying\u0000the geometric equivalence of $k[G]$ which we call circulant coordinate rings.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"74 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192735","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
Quotients of extriangulated categories induced by selforthogonal subcategories 自正交子范畴诱导的外差范畴的四分体
arXiv - MATH - Rings and Algebras Pub Date : 2024-08-26 DOI: arxiv-2408.14098
Peiyu Zhang, Yiwen Shi, Dajun Liu, Li Wang, Jiaqun Wei
{"title":"Quotients of extriangulated categories induced by selforthogonal subcategories","authors":"Peiyu Zhang, Yiwen Shi, Dajun Liu, Li Wang, Jiaqun Wei","doi":"arxiv-2408.14098","DOIUrl":"https://doi.org/arxiv-2408.14098","url":null,"abstract":"Let C be an extriangulated category. We prove that two quotient categories of\u0000extriangu?lated categories induced by selforthogonal subcategories are\u0000equivalent to module categories by restriction of two functors E and Hom,\u0000respectively. Moreover, if the selforthogonal sub?category is contravariantly\u0000finite, then one of the two quotient categories is abelian. This result can be\u0000regarded as a generalization of Demonet-Liu and Zhou-Zhu.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192737","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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