Journal of Pure and Applied Algebra最新文献

{"title":"Essentials dimension of reductive groups via generically free representations","authors":"Sanghoon Baek,&nbsp;Yeongjong Kim","doi":"10.1016/j.jpaa.2025.108044","DOIUrl":"10.1016/j.jpaa.2025.108044","url":null,"abstract":"<div><div>We provide a simple method to compute upper bounds on the essential dimension of split reductive groups with finite or connected center by means of their generically free representations. Combining our upper bound with previously known lower bound, the exact value of the essential dimension is calculated for some types of reductive groups. As an application, we determine the essential dimension of a semisimple group of classical type or <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>6</mn></mrow></msub></math></span>, and its strict reductive envelope under certain conditions on its center. This extends previous works on simple simply connected groups of type <em>B</em> or <em>D</em> by Brosnan-Reichstein-Vistoli and Chernousov-Merkurjev, strict reductive envelopes of groups of type <em>A</em> by Cernele-Reichstein, and semisimple groups of type <em>B</em> by the authors to any classical type and type <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>6</mn></mrow></msub></math></span> in a uniform way.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 9","pages":"Article 108044"},"PeriodicalIF":0.7,"publicationDate":"2025-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144655012","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Principal Pin-bundles and associated vector bundles over Klein surfaces","authors":"Ewa Tyszkowska","doi":"10.1016/j.jpaa.2025.108047","DOIUrl":"10.1016/j.jpaa.2025.108047","url":null,"abstract":"<div><div>We represent a Klein surface <em>Y</em> of algebraic genus <span><math><mi>d</mi><mo>&gt;</mo><mn>1</mn></math></span> as the orbit space of another Klein surface under the action of a multiplicative subgroup of a Clifford algebra. We construct a principal Pin-bundle over <em>Y</em> and a principal Spin-bundle over the canonical double cover <span><math><msup><mrow><mi>Y</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> of <em>Y</em>. We demonstrate how isomorphisms of Clifford algebras affect the associated vector bundles over <em>Y</em> and <span><math><msup><mrow><mi>Y</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 9","pages":"Article 108047"},"PeriodicalIF":0.7,"publicationDate":"2025-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144670922","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Realizing algebra structures on free resolutions of grade 3 perfect ideals","authors":"Alexis Hardesty","doi":"10.1016/j.jpaa.2025.108040","DOIUrl":"10.1016/j.jpaa.2025.108040","url":null,"abstract":"<div><div>Perfect ideals <em>I</em> of grade 3 in a local ring <span><math><mo>(</mo><mi>R</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> can be classified based on multiplicative structures on <span><math><msubsup><mrow><mi>Tor</mi></mrow><mrow><mo>•</mo></mrow><mrow><mi>R</mi></mrow></msubsup><mo>(</mo><mi>R</mi><mo>/</mo><mi>I</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span>. The classification is incomplete in the sense that it remains open which of the possible algebra structures actually occur; this <em>realizability question</em> was formally posed by Avramov in 2012. Of five classes of algebra structures, the realizability question has been answered for one class. In this work, we answer the realizability question for two more classes and contribute towards an answer for a third.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 9","pages":"Article 108040"},"PeriodicalIF":0.7,"publicationDate":"2025-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144580755","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Homogeneous Ulrich bundles on isotropic flag varieties","authors":"Xinyi Fang ,&nbsp;Yusuke Nakayama","doi":"10.1016/j.jpaa.2025.108043","DOIUrl":"10.1016/j.jpaa.2025.108043","url":null,"abstract":"<div><div>In this paper, we consider the existence problem of Ulrich bundles on a rational homogeneous space <span><math><mi>G</mi><mo>/</mo><mi>P</mi></math></span> of type <em>B</em>, <em>C</em> or <em>D</em>. We show that if the Picard number of <span><math><mi>G</mi><mo>/</mo><mi>P</mi></math></span> is greater than or equal to 2, then there are no irreducible homogeneous Ulrich bundles on <span><math><mi>G</mi><mo>/</mo><mi>P</mi></math></span> with respect to the minimal ample class.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 9","pages":"Article 108043"},"PeriodicalIF":0.7,"publicationDate":"2025-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144570798","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Ascent and descent of Gorenstein homological properties","authors":"Jian Liu ,&nbsp;Wei Ren","doi":"10.1016/j.jpaa.2025.108042","DOIUrl":"10.1016/j.jpaa.2025.108042","url":null,"abstract":"<div><div>Let <span><math><mi>φ</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>A</mi></math></span> be a ring homomorphism, where <em>R</em> is a commutative noetherian ring and <em>A</em> is a finite <em>R</em>-algebra. We provide criteria for detecting the ascent and descent of Gorenstein homological properties. Furthermore, we observe that the ascent and descent of Gorenstein homological property can detect the Gorenstein property of rings along <em>φ</em>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 9","pages":"Article 108042"},"PeriodicalIF":0.7,"publicationDate":"2025-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144570797","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Image of ideals under linear K-derivations and the LNED conjecture","authors":"Sakshi Gupta","doi":"10.1016/j.jpaa.2025.108041","DOIUrl":"10.1016/j.jpaa.2025.108041","url":null,"abstract":"<div><div>Let <em>K</em> be a field of characteristic zero and <span><math><mi>K</mi><mo>[</mo><mi>X</mi><mo>]</mo><mo>=</mo><mi>K</mi><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></math></span> be the polynomial algebra in <em>n</em> variables over <em>K</em>. We show that, for a linear <em>K</em>-derivation <em>d</em> of <span><math><mi>K</mi><mo>[</mo><mi>X</mi><mo>]</mo></math></span> and the maximal ideal <span><math><mi>m</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span> of <span><math><mi>K</mi><mo>[</mo><mi>X</mi><mo>]</mo></math></span>, if <span><math><mi>d</mi><mo>(</mo><mi>m</mi><mo>)</mo></math></span> is a Mathieu-Zhao subspace of <span><math><mi>K</mi><mo>[</mo><mi>X</mi><mo>]</mo></math></span>, then the image of every <span><math><mi>m</mi></math></span>-primary ideal under <em>d</em> forms a Mathieu-Zhao subspace of <span><math><mi>K</mi><mo>[</mo><mi>X</mi><mo>]</mo></math></span>. Additionally, we observe that the image of all monomial ideals under the <em>K</em>-derivation <span><math><mi>d</mi><mo>=</mo><mi>f</mi><msub><mrow><mo>∂</mo></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub></math></span> of <span><math><mi>K</mi><mo>[</mo><mi>X</mi><mo>]</mo></math></span>, for <span><math><mi>f</mi><mo>∈</mo><mi>K</mi><mo>[</mo><mi>X</mi><mo>]</mo></math></span> forms an ideal of <span><math><mi>K</mi><mo>[</mo><mi>X</mi><mo>]</mo></math></span>. Finally, we prove that the image of certain monomial ideals under a linear locally nilpotent <em>K</em>-derivation of <span><math><mi>K</mi><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>]</mo></math></span> defined by <span><math><mi>d</mi><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mo>∂</mo></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msub><msub><mrow><mo>∂</mo></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></math></span> forms a Mathieu-Zhao subspace.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 9","pages":"Article 108041"},"PeriodicalIF":0.7,"publicationDate":"2025-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144572879","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Gelfand-Fuks cohomology of vector fields on algebraic varieties","authors":"Yuly Billig,&nbsp;Kathlyn Dykes","doi":"10.1016/j.jpaa.2025.108039","DOIUrl":"10.1016/j.jpaa.2025.108039","url":null,"abstract":"<div><div>For an affine algebraic variety, we introduce algebraic Gelfand-Fuks cohomology of polynomial vector fields with coefficients in differentiable <em>AV</em>-modules. Its complex is given by cochains that are differential operators in the sense of Grothendieck. Using the jets of vector fields, we compute this cohomology for varieties with uniformizing parameters. We prove that in this case, Gelfand-Fuks cohomology with coefficients in a tensor module decomposes as a tensor product of the de Rham cohomology of the variety and the cohomology of the Lie algebra of vector fields on affine space, vanishing at the origin. We explicitly compute this cohomology for affine space, the torus, and Krichever-Novikov algebras.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 9","pages":"Article 108039"},"PeriodicalIF":0.7,"publicationDate":"2025-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144572880","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On dynamical skew braces and skew bracoids","authors":"Davide Ferri","doi":"10.1016/j.jpaa.2025.108036","DOIUrl":"10.1016/j.jpaa.2025.108036","url":null,"abstract":"<div><div>Dynamical skew braces are known to produce solutions to the quiver-theoretic Yang–Baxter equation. Under a technical hypothesis, we prove that these solutions are braided groupoids (and hence skew bracoids in the sense of Sheng, Tang, and Zhu). Conversely, every connected braided groupoid can be <em>parallelised</em>, making it isomorphic to a dynamical skew brace. We study the combinatorics of these objects, depending on some strings of integer invariants.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 9","pages":"Article 108036"},"PeriodicalIF":0.7,"publicationDate":"2025-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144490785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Counting subgroups of a finite group containing a prescribed subgroup","authors":"Lorenzo Guerra ,&nbsp;Fabio Mastrogiacomo ,&nbsp;Pablo Spiga","doi":"10.1016/j.jpaa.2025.108038","DOIUrl":"10.1016/j.jpaa.2025.108038","url":null,"abstract":"<div><div>Let <em>R</em> be a finite group, and let <em>T</em> be a subgroup of <em>R</em>. We show that there are at most<span><span><span><math><mn>7.3722</mn><msup><mrow><mo>[</mo><mi>R</mi><mo>:</mo><mi>T</mi><mo>]</mo></mrow><mrow><mfrac><mrow><msub><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⁡</mo><mo>[</mo><mi>R</mi><mo>:</mo><mi>T</mi><mo>]</mo></mrow><mrow><mn>4</mn></mrow></mfrac><mo>+</mo><mn>1.8919</mn></mrow></msup></math></span></span></span> subgroups of <em>R</em> containing <em>T</em>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 9","pages":"Article 108038"},"PeriodicalIF":0.7,"publicationDate":"2025-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144471326","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Bimodules and universal enveloping algebras associated to SVOAs","authors":"Shun Xu","doi":"10.1016/j.jpaa.2025.108037","DOIUrl":"10.1016/j.jpaa.2025.108037","url":null,"abstract":"&lt;div&gt;&lt;div&gt;For a vertex operator superalgebra &lt;em&gt;V&lt;/em&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, let &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; denote the associative algebra, and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; denote the &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-bimodule, as constructed by W. Jiang and C. Jiang &lt;span&gt;&lt;span&gt;[10]&lt;/span&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; are specific subspaces of &lt;em&gt;V&lt;/em&gt;. We introduce a novel representation-theoretic method for constructing subspaces &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; of &lt;em&gt;V&lt;/em&gt;, similar to our previous work &lt;span&gt;&lt;span&gt;[8]&lt;/span&gt;&lt;/span&gt;, and set &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. We demonstrate that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; through a method that is notably simpler and more straightforward compared to the approach detailed in &lt;span&gt;&lt;span&gt;[6]&lt;/span&gt;&lt;/span&gt; (also see &lt;span&gt;&lt;span&gt;[8]&lt;/span&gt;&lt;/span&gt;). Moreover, we offer a simpler definition for the bimodules &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, contributing towards the resolution of a conjecture proposed by Dong and Jiang &lt;span&gt;&lt;span&gt;[2]&lt;/span&gt;&lt;/span&gt; regarding superalgebras. Additionally, we demonstrate that the &lt;span&gt;&lt;math&gt;&lt;m","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 9","pages":"Article 108037"},"PeriodicalIF":0.7,"publicationDate":"2025-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144471231","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}