International Journal of Algebra and Computation最新文献

{"title":"Count-free Weisfeiler–Leman and group isomorphism","authors":"Nathaniel A. Collins, Michael Levet","doi":"10.1142/s0218196724500103","DOIUrl":"https://doi.org/10.1142/s0218196724500103","url":null,"abstract":"<p>We investigate the power of counting in <span>Group Isomorphism</span>. We first leverage the count-free variant of the Weisfeiler–Leman Version I algorithm for groups [J. Brachter and P. Schweitzer, On the Weisfeiler–Leman dimension of finite groups, in <i>35th Annual ACM/IEEE Symp. Logic in Computer Scienc</i>e, eds. H. Hermanns, L. Zhang, N. Kobayashi and D. Miller, Saarbrucken, Germany, July 8–11, 2020 (ACM, 2020), pp. 287–300, doi:10.1145/3373718.3394786] in tandem with bounded non-determinism and limited counting to improve the parallel complexity of isomorphism testing for several families of groups. These families include:</p><ul><li><p>Direct products of non-Abelian simple groups.</p></li><li><p>Coprime extensions, where the normal Hall subgroup is Abelian and the complement is an <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>O</mi><mo stretchy=\"false\">(</mo><mn>1</mn><mo stretchy=\"false\">)</mo></math></span><span></span>-generated solvable group with solvability class poly log log <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi></math></span><span></span>. This notably includes instances where the complement is an <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>O</mi><mo stretchy=\"false\">(</mo><mn>1</mn><mo stretchy=\"false\">)</mo></math></span><span></span>-generated nilpotent group. This problem was previously known to be in <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi mathvariant=\"sans-serif\">P</mi></math></span><span></span> [Y. Qiao, J. M. N. Sarma and B. Tang, On isomorphism testing of groups with normal Hall subgroups, in <i>Proc. 28th Symp. Theoretical Aspects of Computer Science,</i> Dagstuhl Castle, Leibniz Center for Informatics, 2011), pp. 567–578, doi:10.4230/LIPIcs. STACS.2011.567], and the complexity was recently improved to <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi mathvariant=\"sans-serif\">L</mi></math></span><span></span> [J. A. Grochow and M. Levet, On the parallel complexity of group isomorphism via Weisfeiler–Leman, in <i>24th Int. Symp. Fundamentals of Computation Theory</i>, eds. H. Fernau and K. Jansen, Lecture Notes in Computer Science, Vol. 14292, September 18–21, 2023, Trier, Germany (Springer, 2023), pp. 234–247].</p></li><li><p>Graphical groups of class <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mn>2</mn></math></span><span></span> and exponent <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi><mo>&gt;</mo><mn>2</mn></math></span><span></span> [A. H. Mekler, Stability of nilpotent groups of class 2 and prime exponent, <i>J. Symb. Logic</i><b>46</b>(4) (1981) 781–788] arising from the CFI and twisted CFI graphs [J. -Y. Cai, M. Fürer and N. Immerman, An optimal lower bound on the number of variables for graph identification, <i>Combinatorica</i><b>12</b>(4) (1992) 389–410], respectively. In particular, our work imp","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140563344","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":"Gapsets and the k-generalized Fibonacci sequences","authors":"Gilberto B. Almeida Filho, Matheus Bernardini","doi":"10.1142/s0218196724500085","DOIUrl":"https://doi.org/10.1142/s0218196724500085","url":null,"abstract":"<p>We bring the terminology of the Kunz coordinates of numerical semigroups to gapsets and we generalize this concept to <i>m</i>-extensions. It allows us to identify gapsets and, in general, <i>m</i>-extensions with tilings of boards; as a consequence, we present some applications of this identification. Moreover, we present explicit formulas for the number of gapsets with fixed genus and depth, when the multiplicity is 3 or 4, and, in some cases, for the number of gapsets with fixed genus and depth.</p>","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140597591","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":"Bounding embedded singularities of Hilbert schemes of points on affine three space","authors":"Jen-Chieh Hsiao","doi":"10.1142/s0218196724500140","DOIUrl":"https://doi.org/10.1142/s0218196724500140","url":null,"abstract":"<p>The Hilbert scheme <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mstyle><mtext mathvariant=\"normal\">Hilb</mtext></mstyle></mrow><mrow><mi>n</mi></mrow></msup><msup><mrow><mi>ℂ</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span><span></span> of <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi></math></span><span></span> points on <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>ℂ</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span><span></span> can be expressed as the critical locus of a regular function on a smooth variety <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi mathvariant=\"cal\">𝒳</mi></math></span><span></span>. Recent development in birational geometry suggests a study of singularities of the pair <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mo stretchy=\"false\">(</mo><mi mathvariant=\"cal\">𝒳</mi><mo>,</mo><msup><mrow><mstyle><mtext mathvariant=\"normal\">Hilb</mtext></mstyle></mrow><mrow><mi>n</mi></mrow></msup><msup><mrow><mi>ℂ</mi></mrow><mrow><mn>3</mn></mrow></msup><mo stretchy=\"false\">)</mo></math></span><span></span> using jet schemes. In this paper, we use a comparison between <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mstyle><mtext mathvariant=\"normal\">Hilb</mtext></mstyle></mrow><mrow><mi>n</mi></mrow></msup><msup><mrow><mi>ℂ</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span><span></span> and the scheme <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn><mo>,</mo><mi>n</mi></mrow></msub></math></span><span></span> of three commuting <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi><mo stretchy=\"false\">×</mo><mi>n</mi></math></span><span></span> matrices to estimate the log canonical threshold of <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mo stretchy=\"false\">(</mo><mi mathvariant=\"cal\">𝒳</mi><mo>,</mo><msup><mrow><mstyle><mtext mathvariant=\"normal\">Hilb</mtext></mstyle></mrow><mrow><mi>n</mi></mrow></msup><msup><mrow><mi>ℂ</mi></mrow><mrow><mn>3</mn></mrow></msup><mo stretchy=\"false\">)</mo></math></span><span></span>. As a consequence, we see that although both <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><mo>dim</mo><mi mathvariant=\"cal\">𝒳</mi></math></span><span></span> and <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mo>dim</mo><msup><mrow><mstyle><mtext mathvariant=\"normal\">Hilb</mtext></mstyle></mrow><mrow><mi>n</mi></mrow></msup><msup><mrow><mi>ℂ</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span><span></span> have asymptotic growth <span><math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"><mi>O</mi><mo stretchy=\"false\">(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy=\"false\">)</mo></math></span><span></span>,","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140563786","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 automorphisms of certain free nilpotent-by-abelian Lie algebras","authors":"C. E. Kofinas, A. I. Papistas","doi":"10.1142/s0218196724500097","DOIUrl":"https://doi.org/10.1142/s0218196724500097","url":null,"abstract":"<p>For a positive integer <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi><mo>≥</mo><mn>4</mn></math></span><span></span>, let <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span><span></span> be a free (nilpotent of class 2)-by-abelian and abelian-by-(nilpotent of class 2) Lie algebra of rank <i>n</i>. We show that the subgroup of <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext mathvariant=\"normal\">Aut</mtext></mstyle><mo stretchy=\"false\">(</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">)</mo></math></span><span></span> generated by the tame automorphisms and a countably infinite set of explicitly given automorphisms of <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span><span></span> is dense in <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext mathvariant=\"normal\">Aut</mtext></mstyle><mo stretchy=\"false\">(</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">)</mo></math></span><span></span> with respect to the formal power series topology.</p>","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140597595","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":"Products of traceless and semi-traceless matrices over division rings and their applications","authors":"Peter V. Danchev, Truong Huu Dung, Tran Nam Son","doi":"10.1142/s0218196724500115","DOIUrl":"https://doi.org/10.1142/s0218196724500115","url":null,"abstract":"<p>In this paper, we prove that every matrix over a division ring is representable as a product of at most 10 traceless matrices as well as a product of at most four semi-traceless matrices. By applying this result and the obtained so far other results, we show that elements of some algebras possess some rather interesting and nontrivial decompositions into products of images of non-commutative polynomials.</p>","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140315633","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":"Congruence-simple matrix semirings","authors":"Vítězslav Kala, T. Kepka, M. Korbelář","doi":"10.1142/s0218196724500152","DOIUrl":"https://doi.org/10.1142/s0218196724500152","url":null,"abstract":"","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140238183","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 the structure of finitely presented Bestvina–Brady groups","authors":"Priyavrat Deshpande, Mallika Roy","doi":"10.1142/s0218196724500012","DOIUrl":"https://doi.org/10.1142/s0218196724500012","url":null,"abstract":"<p>Right-angled Artin groups and their subgroups are of great interest because of their geometric, combinatorial and algorithmic properties. It is convenient to define these groups using finite simplicial graphs. The isomorphism type of the group is uniquely determined by the graph. Moreover, many structural properties of right-angled Artin groups can be expressed in terms of their defining graph.</p><p>In this paper, we address the question of understanding the structure of a class of subgroups of right-angled Artin groups in terms of the graph. Bestvina and Brady, in their seminal work, studied these subgroups (now called Bestvina–Brady groups or Artin kernels) from a finiteness conditions viewpoint. Unlike the right-angled Artin groups the isomorphism type of Bestvina–Brady groups is not uniquely determined by the defining graph. We prove that certain finitely presented Bestvina–Brady groups can be expressed as an iterated amalgamated product. Moreover, we show that this amalgamated product can be read off from the graph defining the ambient right-angled Artin group.</p>","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140105993","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":"The omega-reducibility of pseudovarieties of ordered monoids representing low levels of concatenation hierarchies","authors":"Jana Volaříková","doi":"10.1142/s0218196724500024","DOIUrl":"https://doi.org/10.1142/s0218196724500024","url":null,"abstract":"<p>We deal with the question of the <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>ω</mi></math></span><span></span>-reducibility of pseudovarieties of ordered monoids corresponding to levels of concatenation hierarchies of regular languages. A pseudovariety of ordered monoids <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mstyle mathvariant=\"sans-serif\"><mi>V</mi></mstyle></math></span><span></span> is called <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>ω</mi></math></span><span></span>-reducible if, given a finite ordered monoid <i>M</i>, for every inequality of pseudowords that is valid in <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mstyle mathvariant=\"sans-serif\"><mi>V</mi></mstyle></math></span><span></span>, there exists an inequality of <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>ω</mi></math></span><span></span>-words that is also valid in <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mstyle mathvariant=\"sans-serif\"><mi>V</mi></mstyle></math></span><span></span> and has the same “imprint” in <i>M</i>.</p><p>Place and Zeitoun have recently proven the decidability of the membership problem for levels <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mn>1</mn><mo stretchy=\"false\">∕</mo><mn>2</mn></math></span><span></span>, 1, <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mn>3</mn><mo stretchy=\"false\">∕</mo><mn>2</mn></math></span><span></span> and <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mn>5</mn><mo stretchy=\"false\">∕</mo><mn>2</mn></math></span><span></span> of concatenation hierarchies with level 0 being a finite Boolean algebra of regular languages closed under quotients. The solutions of these membership problems have been found by considering a more general problem of separation of regular languages and its further generalization — a problem of covering. Following the results of Place and Zeitoun, we prove that, for every concatenation hierarchy with level 0 being represented by a locally finite pseudovariety of monoids, the pseudovarieties corresponding to levels <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><mn>1</mn><mo stretchy=\"false\">∕</mo><mn>2</mn></math></span><span></span> and <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mn>3</mn><mo stretchy=\"false\">∕</mo><mn>2</mn></math></span><span></span> are <span><math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"><mi>ω</mi></math></span><span></span>-reducible. As a corollary of these results, we obtain that, for every concatenation hierarchy with level 0 being represented by a locally finite pseudovariety of monoids, the pseudovarieties corresponding to levels <span><math altimg=\"eq-00013.gif\" display=\"inline\" overflow=\"scroll\"><mn>3</mn><mo stretchy=\"false\">∕</mo><mn>2</mn></math></span><span></span","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140106337","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":"Commuting and product-zero probability in finite rings","authors":"Pavel Shumyatsky, Matteo Vannacci","doi":"10.1142/s0218196724500061","DOIUrl":"https://doi.org/10.1142/s0218196724500061","url":null,"abstract":"<p>Let <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext mathvariant=\"normal\">cp</mtext></mstyle><mo stretchy=\"false\">(</mo><mi>R</mi><mo stretchy=\"false\">)</mo></math></span><span></span> be the probability that two random elements of a finite ring <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>R</mi></math></span><span></span> commute and <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext mathvariant=\"normal\">zp</mtext></mstyle><mo stretchy=\"false\">(</mo><mi>R</mi><mo stretchy=\"false\">)</mo></math></span><span></span> the probability that the product of two random elements in <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>R</mi></math></span><span></span> is zero. We show that if <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext mathvariant=\"normal\">cp</mtext></mstyle><mo stretchy=\"false\">(</mo><mi>R</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mi>𝜀</mi></math></span><span></span>, then there exists a Lie-ideal <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>D</mi></math></span><span></span> in the Lie-ring <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mo stretchy=\"false\">(</mo><mi>R</mi><mo>,</mo><mo stretchy=\"false\">[</mo><mo stretchy=\"false\">⋅</mo><mo>,</mo><mo stretchy=\"false\">⋅</mo><mo stretchy=\"false\">]</mo><mo stretchy=\"false\">)</mo></math></span><span></span> with <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>𝜀</mi></math></span><span></span>-bounded index and with <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mo stretchy=\"false\">[</mo><mi>D</mi><mo>,</mo><mi>D</mi><mo stretchy=\"false\">]</mo></math></span><span></span> of <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><mi>𝜀</mi></math></span><span></span>-bounded order. If <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext mathvariant=\"normal\">zp</mtext></mstyle><mo stretchy=\"false\">(</mo><mi>R</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mi>𝜀</mi></math></span><span></span>, then there exists an ideal <span><math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"><mi>D</mi></math></span><span></span> in <span><math altimg=\"eq-00013.gif\" display=\"inline\" overflow=\"scroll\"><mi>R</mi></math></span><span></span> with <span><math altimg=\"eq-00014.gif\" display=\"inline\" overflow=\"scroll\"><mi>𝜀</mi></math></span><span></span>-bounded index and <span><math altimg=\"eq-00015.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span><span></span> of <span><math altimg=\"eq-00016.gif\" display=\"inline\" overflow=\"scroll\"><mi>𝜀</mi></math></span><span></span>-bounded order. These results are analogous to the well-known theorem of Neumann on the commuting probability in finite groups.</p>","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140045292","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":"Generalized Gorenstein modules with respect to duality pairs over triangular matrix rings","authors":"Rongmin Zhu, Haiyu Liu","doi":"10.1142/s021819672450005x","DOIUrl":"https://doi.org/10.1142/s021819672450005x","url":null,"abstract":"","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140470909","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}