Pub Date : 2026-01-29DOI: 10.1016/j.jalgebra.2026.01.025
Cătălin Ciupercă
If δ is a derivation on a commutative noetherian ring A containing a field of characteristic zero and k is a positive integer, we study the ideals I of A satisfying . Most results are concerned with the behavior of their integral closures, rational powers, and arbitrary saturations.
{"title":"Ideals invariant under powers of derivations","authors":"Cătălin Ciupercă","doi":"10.1016/j.jalgebra.2026.01.025","DOIUrl":"10.1016/j.jalgebra.2026.01.025","url":null,"abstract":"<div><div>If <em>δ</em> is a derivation on a commutative noetherian ring <em>A</em> containing a field of characteristic zero and <em>k</em> is a positive integer, we study the ideals <em>I</em> of <em>A</em> satisfying <span><math><mi>δ</mi><msup><mrow><mo>(</mo><mi>I</mi><mo>)</mo></mrow><mrow><mi>k</mi></mrow></msup><mo>⊆</mo><mi>I</mi></math></span>. Most results are concerned with the behavior of their integral closures, rational powers, and arbitrary saturations.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"694 ","pages":"Pages 29-43"},"PeriodicalIF":0.8,"publicationDate":"2026-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146098442","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}
Pub Date : 2026-01-29DOI: 10.1016/j.jalgebra.2026.01.023
Toshiyuki Tanisaki
We establish some properties of the ring of differential operators on the quantized flag manifold. Especially, we give an explicit description of its localization on an affine open subset in terms of the quantum Weyl algebra (q-analogue of boson).
{"title":"The ring of differential operators on a quantized flag manifold","authors":"Toshiyuki Tanisaki","doi":"10.1016/j.jalgebra.2026.01.023","DOIUrl":"10.1016/j.jalgebra.2026.01.023","url":null,"abstract":"<div><div>We establish some properties of the ring of differential operators on the quantized flag manifold. Especially, we give an explicit description of its localization on an affine open subset in terms of the quantum Weyl algebra (<em>q</em>-analogue of boson).</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"694 ","pages":"Pages 1-28"},"PeriodicalIF":0.8,"publicationDate":"2026-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146098441","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}
Pub Date : 2026-01-29DOI: 10.1016/j.jalgebra.2025.12.030
Naoki Endo
As part of stratification of Cohen-Macaulay rings, we introduce and develop the theory of Goto rings, generalizing the notion of almost Gorenstein rings originally defined by V. Barucci and R. Fröberg in 1997. What has dominated the series of researches on almost Gorenstein rings is the fact that the reduction numbers of extended canonical ideals are at most 2; we define Goto rings as Cohen-Macaulay rings admitting such extended canonical ideals. We provide a characterization of Goto rings in terms of the structure of Sally modules and determine the Hilbert functions of them. Various examples of Goto rings that come from numerical semigroups, idealizations, fiber products, and equimultiple Ulrich ideals are explored as well.
{"title":"Goto rings","authors":"Naoki Endo","doi":"10.1016/j.jalgebra.2025.12.030","DOIUrl":"10.1016/j.jalgebra.2025.12.030","url":null,"abstract":"<div><div>As part of stratification of Cohen-Macaulay rings, we introduce and develop the theory of Goto rings, generalizing the notion of almost Gorenstein rings originally defined by V. Barucci and R. Fröberg in 1997. What has dominated the series of researches on almost Gorenstein rings is the fact that the reduction numbers of extended canonical ideals are at most 2; we define Goto rings as Cohen-Macaulay rings admitting such extended canonical ideals. We provide a characterization of Goto rings in terms of the structure of Sally modules and determine the Hilbert functions of them. Various examples of Goto rings that come from numerical semigroups, idealizations, fiber products, and equimultiple Ulrich ideals are explored as well.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"694 ","pages":"Pages 44-108"},"PeriodicalIF":0.8,"publicationDate":"2026-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146098461","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}
Pub Date : 2026-01-22DOI: 10.1016/j.jalgebra.2026.01.014
Joris van der Hoeven, Gleb Pogudin
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.
{"title":"Yet another differential shape lemma","authors":"Joris van der Hoeven, 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":"2026-01-22","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}
Pub Date : 2026-01-21DOI: 10.1016/j.jalgebra.2026.01.003
Jinjin Liang, Wen Chen, Erxiao Wang
We develop the basic and new tools for classifying non-side-to-side tilings of the sphere by congruent triangles. Then we prove that, if the triangle has any irrational angle in degree, such tilings are: a sequence of 1-parameter families of triangles each admitting many 2-layer earth map tilings with 2n() tiles, together with rotational modifications for even n; a 1-parameter family of triangles each admitting a unique tiling with 8 tiles; and a sporadic triangle admitting a unique tiling with 16 tiles. Then a scheme is outlined to classify the case with all angles being rational in degree, justified by some known and new examples.
{"title":"Non-side-to-side tilings of the sphere by congruent triangles with an irrational angle","authors":"Jinjin Liang, Wen Chen, Erxiao Wang","doi":"10.1016/j.jalgebra.2026.01.003","DOIUrl":"10.1016/j.jalgebra.2026.01.003","url":null,"abstract":"<div><div>We develop the basic and new tools for classifying non-side-to-side tilings of the sphere by congruent triangles. Then we prove that, if the triangle has any irrational angle in degree, such tilings are: a sequence of 1-parameter families of triangles each admitting many 2-layer earth map tilings with 2<em>n</em>(<span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span>) tiles, together with rotational modifications for even <em>n</em>; a 1-parameter family of triangles each admitting a unique tiling with 8 tiles; and a sporadic triangle admitting a unique tiling with 16 tiles. Then a scheme is outlined to classify the case with all angles being rational in degree, justified by some known and new examples.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 587-610"},"PeriodicalIF":0.8,"publicationDate":"2026-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146075490","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}
Pub Date : 2026-01-21DOI: 10.1016/j.jalgebra.2026.01.020
Sheela Devadas
The global analogue of a Henselian local ring is a Henselian pair: a ring A and an ideal I which satisfy a condition resembling Hensel's lemma regarding lifting coprime factorizations of polynomials over to factorizations over A. The geometric counterpart is the notion of a Henselian scheme, which is an analogue of a tubular neighborhood in algebraic geometry.
In this paper we revisit the foundations of the theory of Henselian schemes. The pathological behavior of quasi-coherent sheaves on Henselian schemes in characteristic 0 makes them poor models for an “algebraic tube” in characteristic 0. We show that such problems do not arise in positive characteristic, and establish good properties for analogues of smooth and étale maps in the general Henselian setting.
{"title":"Henselian schemes in positive characteristic","authors":"Sheela Devadas","doi":"10.1016/j.jalgebra.2026.01.020","DOIUrl":"10.1016/j.jalgebra.2026.01.020","url":null,"abstract":"<div><div>The global analogue of a Henselian local ring is a Henselian pair: a ring <em>A</em> and an ideal <em>I</em> which satisfy a condition resembling Hensel's lemma regarding lifting coprime factorizations of polynomials over <span><math><mi>A</mi><mo>/</mo><mi>I</mi></math></span> to factorizations over <em>A</em>. The geometric counterpart is the notion of a Henselian scheme, which is an analogue of a tubular neighborhood in algebraic geometry.</div><div>In this paper we revisit the foundations of the theory of Henselian schemes. The pathological behavior of quasi-coherent sheaves on Henselian schemes in characteristic 0 makes them poor models for an “algebraic tube” in characteristic 0. We show that such problems do not arise in positive characteristic, and establish good properties for analogues of smooth and étale maps in the general Henselian setting.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 531-586"},"PeriodicalIF":0.8,"publicationDate":"2026-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146025646","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}
Pub Date : 2026-01-21DOI: 10.1016/j.jalgebra.2025.12.023
Filippo Ambrosio , Andrea Santi
Let be a periodically graded semisimple complex Lie algebra. In this note, we give a uniform proof of the recent result by W. de Graaf and H. V. Lê that the hyperplane arrangement determined by the restrictions of the roots of to a Cartan subspace coincides with the hyperplane arrangement of (complex) reflections of the little Weyl group of .
设g= φ i∈Z/mZgi是一个周期渐变的半简单复李代数。在本文中，我们统一证明了W. de Graaf和H. V. Lê最近的结果，即由g的根对Cartan子空间c∧g1的限制所决定的超平面排列与g= i∈Z/mZgi的小Weyl群的（复）反射的超平面排列是一致的。
{"title":"Hyperplane arrangements and Vinberg's θ-groups","authors":"Filippo Ambrosio , Andrea Santi","doi":"10.1016/j.jalgebra.2025.12.023","DOIUrl":"10.1016/j.jalgebra.2025.12.023","url":null,"abstract":"<div><div>Let <span><math><mi>g</mi><mo>=</mo><msub><mrow><mo>⨁</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>Z</mi><mo>/</mo><mi>m</mi><mi>Z</mi></mrow></msub><msub><mrow><mi>g</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> be a periodically graded semisimple complex Lie algebra. In this note, we give a uniform proof of the recent result by W. de Graaf and H. V. Lê that the hyperplane arrangement determined by the restrictions of the roots of <span><math><mi>g</mi></math></span> to a Cartan subspace <span><math><mi>c</mi><mo>⊂</mo><msub><mrow><mi>g</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> coincides with the hyperplane arrangement of (complex) reflections of the little Weyl group of <span><math><mi>g</mi><mo>=</mo><msub><mrow><mo>⨁</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>Z</mi><mo>/</mo><mi>m</mi><mi>Z</mi></mrow></msub><msub><mrow><mi>g</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 770-789"},"PeriodicalIF":0.8,"publicationDate":"2026-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146075559","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}
Pub Date : 2026-01-21DOI: 10.1016/j.jalgebra.2026.01.006
Satyabrat Sahoo
Let K be a totally real number field and be the ring of integers of K. In this article, we study the asymptotic solutions of the generalized Fermat equation, namely over K with prime exponent p, where with ABC is even. For certain class of fields K, we prove that the equation has no asymptotic solution with . Then, under some assumptions on , we also prove that has no asymptotic solution in . Finally, we give several purely local criteria of K such that has no asymptotic solutions in , and calculate the density of such fields K when K is a real quadratic field.
{"title":"On the solutions of the generalized Fermat equation over totally real number fields","authors":"Satyabrat Sahoo","doi":"10.1016/j.jalgebra.2026.01.006","DOIUrl":"10.1016/j.jalgebra.2026.01.006","url":null,"abstract":"<div><div>Let <em>K</em> be a totally real number field and <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>K</mi></mrow></msub></math></span> be the ring of integers of <em>K</em>. In this article, we study the asymptotic solutions of the generalized Fermat equation, namely <span><math><mi>A</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><mi>B</mi><msup><mrow><mi>y</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><mi>C</mi><msup><mrow><mi>z</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>=</mo><mn>0</mn></math></span> over <em>K</em> with prime exponent <em>p</em>, where <span><math><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>∈</mo><msub><mrow><mi>O</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>∖</mo><mo>{</mo><mn>0</mn><mo>}</mo></math></span> with <em>ABC</em> is even. For certain class of fields <em>K</em>, we prove that the equation <span><math><mi>A</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><mi>B</mi><msup><mrow><mi>y</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><mi>C</mi><msup><mrow><mi>z</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>=</mo><mn>0</mn></math></span> has no asymptotic solution <span><math><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo><mo>∈</mo><msubsup><mrow><mi>O</mi></mrow><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msubsup></math></span> with <span><math><mn>2</mn><mo>|</mo><mi>a</mi><mi>b</mi><mi>c</mi></math></span>. Then, under some assumptions on <span><math><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi></math></span>, we also prove that <span><math><mi>A</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><mi>B</mi><msup><mrow><mi>y</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><mi>C</mi><msup><mrow><mi>z</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>=</mo><mn>0</mn></math></span> has no asymptotic solution in <span><math><msup><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. Finally, we give several purely local criteria of <em>K</em> such that <span><math><mi>A</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><mi>B</mi><msup><mrow><mi>y</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><mi>C</mi><msup><mrow><mi>z</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>=</mo><mn>0</mn></math></span> has no asymptotic solutions in <span><math><msup><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>, and calculate the density of such fields <em>K</em> when <em>K</em> is a real quadratic field.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 690-709"},"PeriodicalIF":0.8,"publicationDate":"2026-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146075493","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}
Pub Date : 2026-01-20DOI: 10.1016/j.jalgebra.2026.01.019
Tommy Hofmann
We consider the computational problem of determining the unit group of a finite ring, by which we mean the computation of a finite presentation together with an algorithm to express units as words in the generators. We show that the problem is equivalent to the number theoretic problems of factoring integers and solving discrete logarithms in finite fields. A similar equivalence is shown for the problem of determining the abelianization of the unit group or the first K-group of finite rings.
{"title":"Determining unit groups and K1 of finite rings","authors":"Tommy Hofmann","doi":"10.1016/j.jalgebra.2026.01.019","DOIUrl":"10.1016/j.jalgebra.2026.01.019","url":null,"abstract":"<div><div>We consider the computational problem of determining the unit group of a finite ring, by which we mean the computation of a finite presentation together with an algorithm to express units as words in the generators. We show that the problem is equivalent to the number theoretic problems of factoring integers and solving discrete logarithms in finite fields. A similar equivalence is shown for the problem of determining the abelianization of the unit group or the first <em>K</em>-group of finite rings.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 510-530"},"PeriodicalIF":0.8,"publicationDate":"2026-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146025708","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}
Pub Date : 2026-01-19DOI: 10.1016/j.jalgebra.2026.01.008
Elshani Kamberi, Francesco Navarra, Ayesha Asloob Qureshi
In this article, we study the squarefree powers of facet ideals associated with simplicial trees. Specifically, we examine the linearity of their minimal free resolution and their regularity. Additionally, we investigate when the first syzygy module of squarefree powers of facet ideal of a simplicial tree is generated by linear relations. Finally, we provide a combinatorial formula for the regularity of the squarefree powers of t-path ideals of path graphs.
{"title":"On squarefree powers of simplicial trees","authors":"Elshani Kamberi, Francesco Navarra, Ayesha Asloob Qureshi","doi":"10.1016/j.jalgebra.2026.01.008","DOIUrl":"10.1016/j.jalgebra.2026.01.008","url":null,"abstract":"<div><div>In this article, we study the squarefree powers of facet ideals associated with simplicial trees. Specifically, we examine the linearity of their minimal free resolution and their regularity. Additionally, we investigate when the first syzygy module of squarefree powers of facet ideal of a simplicial tree is generated by linear relations. Finally, we provide a combinatorial formula for the regularity of the squarefree powers of <em>t</em>-path ideals of path graphs.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 240-276"},"PeriodicalIF":0.8,"publicationDate":"2026-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146025652","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}
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