Journal of Pure and Applied Algebra最新文献

{"title":"M-ideals: from Banach spaces to rings","authors":"David P. Blecher,&nbsp;Amartya Goswami","doi":"10.1016/j.jpaa.2026.108389","DOIUrl":"10.1016/j.jpaa.2026.108389","url":null,"abstract":"<div><div>We introduce and investigate a class of ring ideals, termed ring M<em>-ideals</em>, inspired by the Alfsen–Effros theory of M-ideals in Banach spaces. We show that M-ideals extend the classical notion of essential ideals and subsume them as a subclass. The central theorem provides a full characterization: an ideal is an M-ideal if and only if it is either essential or relatively irreducible. This dichotomy reveals the abundant and diverse nature of M-ideals, encompassing both essential and minimal ideals, and admits natural generalizations in rings beyond the commutative and unital settings.</div><div>We systematically study the algebraic stability of M-ideals under standard constructions such as intersection, quotients by direct-summand ideals, direct product, and Morita equivalence and establish their behavior in various subclasses of rings, such as operator algebras. In certain rings such as <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and C*-algebras, we completely classify M-ideals and relate them to algebraically minimal projections and central idempotents. The ring M-ideals in <span><math><mi>C</mi><mo>(</mo><mi>K</mi><mo>)</mo></math></span> are shown to be precisely the essential ideals or those minimal ideals corresponding to isolated points.</div><div>Structurally, for unital rings we show that the absence of proper M-ideals characterizes simplicity, while rings in which every proper M-ideal is a direct summand must decompose as finite direct sums of simple rings. In closing, we introduce the notion of M-complements, drawing an analogy with essential extensions in module theory, and demonstrate their existence.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 9","pages":"Article 108389"},"PeriodicalIF":0.7,"publicationDate":"2026-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148934309","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":"Monomial and binomial retracts of polynomial rings","authors":"Sagnik Chakraborty,&nbsp;Madhuparna Pal","doi":"10.1016/j.jpaa.2026.108387","DOIUrl":"10.1016/j.jpaa.2026.108387","url":null,"abstract":"<div><div>Let <em>R</em> be a ring and <span><math><mi>B</mi><mo>≔</mo><mi>R</mi><mo>[</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</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> the polynomial ring in <em>n</em> variables over <em>R</em>. In this article, we consider retractions <span><math><mi>φ</mi><mo>:</mo><mi>B</mi><mo>⟶</mo><mi>B</mi></math></span> such that <span><math><mi>φ</mi><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></math></span> is 0, or a monomial or the sum of two monomials. We prove that, under certain conditions on the base ring <em>R</em>, the resulting retracts are also polynomial rings over <em>R</em>. We characterize different monomial retractions of <em>B</em>, i.e., where <span><math><mi>φ</mi><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></math></span> is 0 or a monomial for all <em>i</em>, giving the same retract. We also generalize the structure of monomial retracts by allowing a few variables to map to arbitrary polynomials, but only under some additional restrictions on the base ring <em>R</em>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 9","pages":"Article 108387"},"PeriodicalIF":0.7,"publicationDate":"2026-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148934310","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":"Correction of Theorem 3.8 in “Stability results for projective modules over Rees algebras” [J. Pure Appl. Algebra 223 (1) (2019) 1–9]","authors":"Ravi A. Rao ,&nbsp;Husney Parvez Sarwar","doi":"10.1016/j.jpaa.2026.108253","DOIUrl":"10.1016/j.jpaa.2026.108253","url":null,"abstract":"<div><div>The proof of <span><span>[5, Lemma 3.7]</span></span> is incomplete. The authors think that it may not be true in the generality. This lemma is only used in the proof of <span><span>[5, Theorem 3.8]</span></span>. However, <span><span>[5, Theorem 3.8]</span></span> is correct. Here we provide a different proof which is similar to the earlier proof but avoids the use of <span><span>[5, Lemma 3.7]</span></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 5","pages":"Article 108253"},"PeriodicalIF":0.8,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147996352","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":"Torsion modules and differential operators in infinitely many variables","authors":"Leonid Positselski","doi":"10.1016/j.jpaa.2026.108240","DOIUrl":"10.1016/j.jpaa.2026.108240","url":null,"abstract":"<div><div>This paper grew out of the author's work on <span><span>[17]</span></span>. Differential operators in the sense of Grothendieck acting between modules over a commutative ring can be interpreted as torsion elements in the bimodule of all operators with respect to the diagonal ideal in the tensor square of the ring. Various notions of torsion modules for an infinitely generated ideal in a commutative ring lead to various notions of differential operators. We discuss differential operators of transfinite orders and differential operators having no global order at all, but only local orders with respect to specific elements of the ring. Many examples are presented. In particular, we prove that every ordinal can be realized as the order of a differential operator acting on the algebra of polynomials in infinitely many variables over a field. We also discuss extension of differential operators to localizations of rings and modules, and to colocalizations of modules.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 4","pages":"Article 108240"},"PeriodicalIF":0.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147602051","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":"Irreducibility and singularities of some nested Quot schemes","authors":"Parvez Rasul ,&nbsp;Ronnie Sebastian","doi":"10.1016/j.jpaa.2026.108237","DOIUrl":"10.1016/j.jpaa.2026.108237","url":null,"abstract":"<div><div>Let <em>C</em> be a smooth projective curve over <span><math><mi>C</mi></math></span> of genus <span><math><mi>g</mi><mo>⩾</mo><mn>1</mn></math></span>. Let <em>E</em> be a vector bundle on <em>C</em> of rank <em>r</em> and degree <em>e</em>. Given integers <span><math><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> such that <span><math><mi>r</mi><mo>&gt;</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>&gt;</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>&gt;</mo><mn>0</mn></math></span>, let <span><math><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>E</mi><mo>)</mo></math></span> denote the nested Quot scheme which parametrizes pair of quotients <span><math><mo>[</mo><mi>E</mi><mo>↠</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>↠</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>]</mo></math></span> such that <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> has rank <span><math><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> and degree <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>. We show that these nested Quot schemes are integral, local complete intersection schemes when <span><math><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≫</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>≫</mo><mn>0</mn></math></span> or <span><math><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>≫</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≫</mo><mn>0</mn></math></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 4","pages":"Article 108237"},"PeriodicalIF":0.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147602054","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":"Translational hulls of semigroups of endomorphisms of an algebra","authors":"Victoria Gould ,&nbsp;Ambroise Grau ,&nbsp;Marianne Johnson ,&nbsp;Mark Kambites","doi":"10.1016/j.jpaa.2026.108203","DOIUrl":"10.1016/j.jpaa.2026.108203","url":null,"abstract":"<div><div>We consider the translational hull <span><math><mi>Ω</mi><mo>(</mo><mi>I</mi><mo>)</mo></math></span> of an arbitrary subsemigroup <em>I</em> of an endomorphism monoid <span><math><mi>End</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> where <em>A</em> is a universal algebra. We give conditions for every bi-translation of <em>I</em> to be realised by transformations, or by endomorphisms, of <em>A</em>. We demonstrate that certain of these conditions are also sufficient to provide natural isomorphisms between the translational hull of <em>I</em> and the idealiser of <em>I</em> within <span><math><mi>End</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, which in the case where <em>I</em> is an ideal is simply <span><math><mi>End</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span>. We describe the connection between these conditions and work of Petrich and Gluskin in the context of densely embedded ideals. Where the conditions fail, we develop a methodology to extract information concerning <span><math><mi>Ω</mi><mo>(</mo><mi>I</mi><mo>)</mo></math></span> from the translational hull <span><math><mi>Ω</mi><mo>(</mo><mi>I</mi><mo>/</mo><mo>≈</mo><mo>)</mo></math></span> of a quotient <span><math><mi>I</mi><mo>/</mo><mo>≈</mo></math></span> of <em>I</em>. We illustrate these concepts in detail in the cases where <em>A</em> is: a free algebra; an independence algebra; a finite symmetric group.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 4","pages":"Article 108203"},"PeriodicalIF":0.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147388388","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":"Invariant symmetrizing forms on symmetric cellular algebras","authors":"Ming Fang ,&nbsp;Hongpeng Zhang","doi":"10.1016/j.jpaa.2026.108236","DOIUrl":"10.1016/j.jpaa.2026.108236","url":null,"abstract":"<div><div>A symmetric cellular algebra has an involution from its cellular structure and also a symmetrizing form which is not necessarily invariant under the involution, a priori. This article studies the existence of idempotents and bilinear forms fixed by involutions in a broader context and confirms as an application that any symmetric cellular algebra over a field of characteristic different from two always has an invariant symmetrizing form.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 4","pages":"Article 108236"},"PeriodicalIF":0.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147602049","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":"Pretorsion theories in prenormal categories","authors":"Sandra Mantovani,&nbsp;Mariano Messora","doi":"10.1016/j.jpaa.2026.108239","DOIUrl":"10.1016/j.jpaa.2026.108239","url":null,"abstract":"<div><div>In this paper we extend several classical results on pointed torsion theories – also known as torsion pairs – to the setting of non-pointed torsion theories defined via kernels and cokernels relative to a fixed class of trivial objects (often referred to as <em>pretorsion theories</em>). Our results are developed in the recently introduced framework of (non-pointed) <em>prenormal categories</em> and other related contexts. Within these settings, we recover some characterisations of torsion and torsion-free subcategories, as well as the classical correspondences between torsion theories and closure operators. We also suitably extend a correspondence between torsion theories and (stable) factorisation systems on the ambient category, known in the homological case. Some of these results are then further specialised to an appropriate notion of hereditary torsion theory. Finally, we apply the developed theory to construct new examples of pretorsion theories.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 4","pages":"Article 108239"},"PeriodicalIF":0.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147602053","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":"Homotopy limits and homotopy colimits of chain complexes","authors":"Kensuke Arakawa","doi":"10.1016/j.jpaa.2026.108232","DOIUrl":"10.1016/j.jpaa.2026.108232","url":null,"abstract":"<div><div>We give a formula for homotopy limits and homotopy colimits of diagrams of chain complexes using the cobar and bar constructions, also known as the Bousfield–Kan formula. Along the way, we show that the Bousfield–Kan formula computes homotopy colimits in any framed model category.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 4","pages":"Article 108232"},"PeriodicalIF":0.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147602052","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":"Weakening relations over duals of limit ordinals","authors":"Nikolaos Galatos,&nbsp;Isis A. Gallardo","doi":"10.1016/j.jpaa.2026.108235","DOIUrl":"10.1016/j.jpaa.2026.108235","url":null,"abstract":"<div><div>We prove that all varieties generated by weakening relation algebras over duals of limit ordinals have a decidable equational theory. We also show that there are only countably-many such varieties and they form a chain that embeds in <span><math><msup><mrow><mi>ω</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span>. The time warp algebra is isomorphic to one of these weakening relation algebras (over the dual of the naturals), so we obtain the main result of a recent publication as a special case.</div><div>The algebras we study connect to the theory of relation algebras, to time warps and graded modalities, and to the algebraic semantics of substructural logics. Our methods are inspired by the theory of lattice-ordered groups (and lattice-ordered pregroups).</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 4","pages":"Article 108235"},"PeriodicalIF":0.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147602047","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}