Journal of Algebra最新文献

{"title":"F-thresholds of filtrations of ideals","authors":"Mitra Koley ,&nbsp;Arvind Kumar","doi":"10.1016/j.jalgebra.2026.01.012","DOIUrl":"10.1016/j.jalgebra.2026.01.012","url":null,"abstract":"<div><div>In this article, we extend the notion of <em>F</em>-thresholds of ideals to <em>F</em>-thresholds of filtrations of ideals. We establish the existence of <em>F</em>-thresholds for various types of filtrations, including symbolic power filtrations and integral closure filtrations. Additionally, we outline various necessary and sufficient conditions for the finiteness of <em>F</em>-thresholds. We also provide effective upper bounds for symbolic <em>F</em>-thresholds.</div><div>Furthermore, we provide a numerical criterion for the symbolic <em>F</em>-splitting, a recently defined <em>F</em>-singularity, in terms of its symbolic <em>F</em>-threshold. We initiate the study of <em>F</em>-thresholds through valuation theory and establish an upper bound in terms of valuations. We also present a formula for the <em>F</em>-threshold of any monomial ideal in terms of its Rees valuations. Lastly, we compute symbolic <em>F</em>-thresholds for various determinantal ideals, <em>F</em>-König ideals, and more.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 1-35"},"PeriodicalIF":0.8,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146001819","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 generalized Gorenstein local rings","authors":"Shiro Goto ,&nbsp;Shinya Kumashiro","doi":"10.1016/j.jalgebra.2026.01.011","DOIUrl":"10.1016/j.jalgebra.2026.01.011","url":null,"abstract":"<div><div>In this paper, we introduce generalized Gorenstein local (GGL) rings. The notion of GGL rings is a natural generalization of the notion of almost Gorenstein rings, which can thus be treated as part of the theory of GGL rings. For a Cohen-Macaulay local ring <em>R</em>, we explore the endomorphism algebra of the maximal ideal, the trace ideal of the canonical module, Ulrich ideals, and Rees algebras of parameter ideals in connection with the GGL property. We also give numerous examples of numerical semigroup rings, idealizations, and determinantal rings of certain matrices.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 98-156"},"PeriodicalIF":0.8,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146025649","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":"Yet another differential shape lemma","authors":"Joris van der Hoeven,&nbsp;Gleb Pogudin","doi":"10.1016/j.jalgebra.2026.01.014","DOIUrl":"10.1016/j.jalgebra.2026.01.014","url":null,"abstract":"<div><div>Recently, Kauers, Koutschan, and Verron proved a non-commutative version of the classical shape lemma in the theory of Gröbner bases. Their result requires the ideal to be D-radical. In this note, we prove a new non-commutative shape lemma that does not require this assumption.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 683-689"},"PeriodicalIF":0.8,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146075488","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":"Affinization of algebraic structures: Leibniz algebras","authors":"Tomasz Brzeziński ,&nbsp;Krzysztof Radziszewski ,&nbsp;Brais Ramos Pérez","doi":"10.1016/j.jalgebra.2026.01.018","DOIUrl":"10.1016/j.jalgebra.2026.01.018","url":null,"abstract":"<div><div>A general procedure of affinization of linear algebra structures is illustrated by the case of Leibniz algebras. Specifically, the definition of an affine Leibniz bracket, that is, a bi-affine operation on an affine space that at each tangent vector space becomes a (bi-linear) Leibniz bracket in terms of a tri-affine operation called a Leibnizian, is given. An affine space together with such an operation is called a Leibniz affgebra. It is shown that any Leibniz algebra can be extended to a family of Leibniz affgebras. Depending on the choice of a Leibnizian different types of Leibniz affgebras are introduced. These include: derivative-type, which captures the derivation property of linear Leibniz bracket; homogeneous-type, which is based on the simplest and least restrictive choice of the Leibnizian; Lie-type which includes all Lie affgebras introduced in R.R. Andruszkiewicz, T. Brzeziński &amp; K. Radziszewski (2025) <span><span>[1]</span></span>. Each type is illustrated by examples with prescribed Leibniz algebra fibres.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 329-361"},"PeriodicalIF":0.8,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146025654","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":"Galois subspaces for projective varieties","authors":"Robert Auffarth","doi":"10.1016/j.jalgebra.2026.01.007","DOIUrl":"10.1016/j.jalgebra.2026.01.007","url":null,"abstract":"<div><div>Given an embedding of a projective variety into projective space, we study the structure of the space of all linear projections that, when composed with the embedding, give a Galois morphism from the variety to a projective space of the same dimension.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 62-77"},"PeriodicalIF":0.8,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146001822","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":"Khovanskii bases of subalgebras arising from finite distributive lattices","authors":"Akihiro Higashitani,&nbsp;Koji Matsushita ,&nbsp;Koichiro Tani","doi":"10.1016/j.jalgebra.2025.12.029","DOIUrl":"10.1016/j.jalgebra.2025.12.029","url":null,"abstract":"<div><div>The notion of Khovanskii bases was introduced by Kaveh and Manon <span><span>[6]</span></span>. It is a generalization of the notion of SAGBI bases for a subalgebra of polynomials. The notion of SAGBI bases was introduced by Robbiano and Sweedler <span><span>[10]</span></span> as an analogue of Gröbner bases in the context of subalgebras. A Hibi ideal is an ideal of a polynomial ring that arises from a distributive lattice. For the development of an analogy of the theory of Hibi ideals and Gröbner bases within the framework of subalgebras, in this paper, we investigate when the set of the polynomials associated with a distributive lattice forms a Khovanskii basis of the subalgebras it generates. We characterize such distributive lattices and their underlying posets. In particular, generalized snake posets and <span><math><mo>{</mo><mo>(</mo><mn>2</mn><mo>+</mo><mn>2</mn><mo>)</mo><mo>,</mo><mo>(</mo><mn>1</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>1</mn><mo>)</mo><mo>}</mo></math></span>-free posets appear as the characterization.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 805-823"},"PeriodicalIF":0.8,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146170685","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 lattice of the weak factorization systems on a finite lattice","authors":"Yongle Luo ,&nbsp;Baptiste Rognerud","doi":"10.1016/j.jalgebra.2026.01.017","DOIUrl":"10.1016/j.jalgebra.2026.01.017","url":null,"abstract":"<div><div>For a finite lattice, we consider the lattice of all its weak factorization systems, or equivalently of all its transfer systems. We prove that it enjoys very strong properties such as semidistributivity, trimness and congruence uniformity. We introduce the elevating graph of a finite lattice as a particular graph whose vertices are the relations of the lattice and we prove that there is a bijection between the transfer systems and the cliques of this graph. This bijection provides a combinatorial model for the problem of enumerating the transfer systems. As illustrations, we recover a known result for the diamond lattices and we obtain a very large lower bound for the boolean lattices.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 277-328"},"PeriodicalIF":0.8,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146025653","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":"Nakayama automorphisms of graded double Ore extensions of Koszul Artin-Schelter regular algebras with nontrivial skew derivations","authors":"Yan Cao ,&nbsp;Yuan Shen ,&nbsp;Xin Wang","doi":"10.1016/j.jalgebra.2026.01.009","DOIUrl":"10.1016/j.jalgebra.2026.01.009","url":null,"abstract":"<div><div>Let <em>A</em> be a Koszul Artin-Schelter regular algebra and <span><math><mi>B</mi><mo>=</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>P</mi></mrow></msub><mo>[</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>;</mo><mi>ς</mi><mo>,</mo><mi>ν</mi><mo>]</mo></math></span> be a graded double Ore extension of <em>A</em> where <span><math><mi>ς</mi><mo>:</mo><mi>A</mi><mo>→</mo><msub><mrow><mi>M</mi></mrow><mrow><mn>2</mn><mo>×</mo><mn>2</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span> is a graded algebra homomorphism and <span><math><mi>ν</mi><mo>:</mo><mi>A</mi><mo>→</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>⊕</mo><mn>2</mn></mrow></msup></math></span> is a degree one <em>ς</em>-derivation. We construct a minimal free resolution for the trivial module of <em>B</em>, and it implies that <em>B</em> is still Koszul. We introduce a homological invariant called <em>ς</em>-divergence of <em>ν</em>, and with its aid, we obtain a precise description of the Nakayama automorphism of <em>B</em>. A twisted superpotential <span><math><mover><mrow><mi>ω</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span> for <em>B</em> with respect to the Nakayama automorphism is constructed so that <em>B</em> is isomorphic to the derivation quotient algebra of <span><math><mover><mrow><mi>ω</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 403-449"},"PeriodicalIF":0.8,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146025648","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 motivic homotopy type of algebraic stacks","authors":"Neeraj Deshmukh ,&nbsp;Jack Hall","doi":"10.1016/j.jalgebra.2026.01.005","DOIUrl":"10.1016/j.jalgebra.2026.01.005","url":null,"abstract":"<div><div>We construct smooth presentations of algebraic stacks that are local epimorphisms in the Morel–Voevodsky <span><math><msup><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-homotopy category. As a consequence, we show that the motive of a smooth stack (in Voevodsky's triangulated category of motives) has many of the same properties as the motive of a smooth scheme.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 790-804"},"PeriodicalIF":0.8,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146075489","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":"A note on a conjecture of Rossi for reduction numbers of ideals and their Ratliff-Rush filtration","authors":"Anoot Kumar Yadav ,&nbsp;Kumari Saloni","doi":"10.1016/j.jalgebra.2025.12.024","DOIUrl":"10.1016/j.jalgebra.2025.12.024","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; be a Cohen-Macaulay local ring of dimension &lt;span&gt;&lt;math&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; and &lt;em&gt;I&lt;/em&gt; an &lt;span&gt;&lt;math&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-primary ideal. In this paper, we prove that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; if &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;I&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;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; for &lt;span&gt;&lt;math&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; where &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is the reduction number of &lt;em&gt;I&lt;/em&gt; with respect to &lt;em&gt;J&lt;/em&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; are the Hilbert coefficients. Our result affirms a conjecture of M.E. Rossi. We also prove that (i) &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;I&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;/math&gt;&lt;/span&gt; for any &lt;em&gt;I&lt;/em&gt;-admissible filtration &lt;span&gt;&lt;math&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and (ii) &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for an integrally closed ideal &lt;em&gt;I&lt;/em&gt;. The above bound for &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; in the case of &lt;span&gt;&lt;math&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-primary ideals is better than the earlier known bounds in our knowledge. Further, the respective","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 213-239"},"PeriodicalIF":0.8,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146025651","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}