Let $R$ be a commutative unital ring and $N$ be a submodule of an $R$-module $M$. The submodule $langle E_M(N)rangle$ generated by the envelope $E_M(N)$ of $N$ is instrumental in studying rings and modules that satisfy the radical formula. We show that: 1) the semiprime radical is an invariant on all the submodules which are respectively generated by envelopes in the ascending chain of envelopes of a given submodule; 2) for rings that satisfy the radical formula, $langle E_M(0)rangle$ is an idempotent radical and it induces a torsion theory whose torsion class consists of all nil $R$-modules and the torsionfree class consists of all reduced $R$-modules; 3) Noetherian uniserial modules satisfy the semiprime radical formula and their semiprime radical is a nil module; and lastly, 4) we construct a sheaf of nil $R$-modules on $text{Spec}(R)$.
{"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}
In this paper, we present the PBW-deformations of the smash product $A # H_{2n^2}$ and the braided tensor product $(A otimes^c A^{mathrm{op}}_c) # H_{2n^2}$ for the Kac-Paljutkin type Hopf algebra $H_{2n^2}$ and its Koszul $H_{2n^2}$-module algebra $A$. A necessary and sufficient condition for the Koszul dual algebra $A^{!}$ to be an $H$-module algebra is given. Then we provide all the PBW-deformations of the smash product algebra $A^! # H_{2n^2}$ for a given $H_{2n^2}$-module algebra $A$.
{"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}
We verify the conjectures due to Lewis, Reiner and Stanton about the Hilbert series of the invariant ring of the truncated polynomial ring for all parabolic subgroups up to rank $3$. This is done by constructing an explicit set of generators for each invariant ring in question. We also propose a conjecture concerning the action of the Steenrod algebra and the Dickson algebra on a certain naturally occurring filtration of the invariant ring under the general linear group.
{"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}
Bruns and Roddy constructed a $3$-generated modular ortholattice $L$ which cannot be embedded into any complete modular ortholattice. Motivated by their approach, we use shift operators to construct a $*$-regular $*$-ring $R$ of endomorphisms of an inner product space (which can be chosen as the Hilbert space $ell^2$) such that direct finiteness fails for $R$.
{"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}
Let $F$ be the function field of a smooth, geometrically integral curve over a $p$-adic field with $pneq 2.$ Let $G$ be a classical adjoint group of type $^1D_n$ defined over $F$. We show that $G(F) / R$ is trivial, where $R$ denotes {it rational equivalence} on $G(F)$.
{"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}
Let $X$ be a normal projective variety of dimension $d$ over an algebraically closed field and $f$ an automorphism of $X$. Suppose that the pullback $f^*|_{mathsf{N}^1(X)_mathbf{R}}$ of $f$ on the real N'eron--Severi space $mathsf{N}^1(X)_mathbf{R}$ is unipotent and denote the index of the eigenvalue $1$ by $k+1$. We prove an upper bound for the polynomial volume growth $mathrm{plov}(f)$ of $f$ as follows: [ mathrm{plov}(f) le (k/2 + 1)d. ] This inequality is optimal in certain cases. Furthermore, we show that $kle 2(d-1)$, extending a result of Dinh--Lin--Oguiso--Zhang for compact K"ahler manifolds to arbitrary characteristic. Combining these two inequalities together, we obtain an optimal inequality that [ mathrm{plov}(f) le d^2, ] which affirmatively answers questions of Cantat--Paris-Romaskevich and Lin--Oguiso--Zhang.
{"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}
The ABC conjecture implies many conjectures and theorems in number theory, including the celebrated Fermat's Last Theorem. Mason-Stothers Theorem is a function field analogue of the ABC conjecture that admits a much more elementary proof with many interesting consequences, including a polynomial version of Fermat's Last Theorem. While years of dedicated effort are expected for a full formalization of Fermat's Last Theorem, the simple proof of Mason-Stothers Theorem and its corollaries calls for an immediate formalization. We formalize an elementary proof of by Snyder in Lean 4, and also formalize many consequences of Mason-Stothers, including (i) non-solvability of Fermat-Cartan equations in polynomials, (ii) non-parametrizability of a certain elliptic curve, and (iii) Davenport's Theorem. We compare our work to existing formalizations of Mason-Stothers by Eberl in Isabelle and Wagemaker in Lean 3 respectively. Our formalization is based on the mathlib4 library of Lean 4, and is currently being ported back to mathlib4.
{"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}
We compute 1/2-derivations on the extended Schr"odinger-Virasoro algebras and the original deformative Schr"odinger-Virasoro algebras. The extended Schr"odinger-Virasoro algebras have neither nontrivial 1/2-derivations nor nontrivial transposed Poisson algebra structures. We demonstrate that the original deformative Schr"odinger-Virasoro algebras have nontrivial 1/2-derivations, indicating that they possess nontrivial transposed Poisson structures.
{"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}
We provide the Krull-Remak-Schmidt decomposition of group algebras of the form $k[G]$ where $k$ is a field, which includes fields with prime characteristic, and $G$ a finite abelian group. We achieved this by studying the geometric equivalence of $k[G]$ which we call circulant coordinate rings.
{"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}
Peiyu Zhang, Yiwen Shi, Dajun Liu, Li Wang, Jiaqun Wei
Let C be an extriangulated category. We prove that two quotient categories of extriangu?lated categories induced by selforthogonal subcategories are equivalent to module categories by restriction of two functors E and Hom, respectively. Moreover, if the selforthogonal sub?category is contravariantly finite, then one of the two quotient categories is abelian. This result can be regarded as a generalization of Demonet-Liu and Zhou-Zhu.
让 C 是一个外差范畴。我们证明，由自正交子范畴诱导的外差范畴的两个商范畴，通过两个函数 E 和 Hom 的限制，分别等价于模块范畴。此外，如果自正交子范畴是逆变无限的，那么两个商范畴中就有一个是无性的。这一结果可以看作是刘德莫内和朱周的概括。
{"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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：