On the simplicity of semi-graded rings by hypercentral groups

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dc.contributor.advisorReyes Villamil, Milton Armando
dc.contributor.authorArana Hernández, Daniel Camilo
dc.date.accessioned2025-09-10T16:07:41Z
dc.date.available2025-09-10T16:07:41Z
dc.date.issued2025-04-01
dc.description.abstractEn un artículo publicado por Jespers en 1993, se estableció una caracterización de la simplicidad de anillos graduados por grupos hipercentrales en términos de su graduación y el centro del anillo. En el año 2020 este resultado fue generalizado por Nystedt and Öinert para anillos no asociativos. Por otro lado, en 2017 Lezama y Latorre introdujeron los anillos semi-graduados, los cuales son una familia de estructuras algebraicas que extienden a los anillos graduados y además han sido una herramienta clave para el estudio de cierto tipo de anillos no conmutativos. Dado que los grupos hipercentrales generalizan a los grupos nilpotentes, y estos a su vez contienen a los grupos abelianos, nos preguntamos si existe un resultado similar al de Jespers pero en el ámbito semigraduado, es decir, si es posible caracterizar la simplicidad de este tipo de anillos en términos de una simplicidad semi-graduada. Este trabajo busca estudiar, explorar y formular preguntas y respuestas en esta dirección. (Texto tomado de la fuente).spa
dc.description.abstractIn a paper published by Jespers in 1993, a characterization of the simplicity of rings graded by hypercentral groups was established in terms of their graduation and the center of the ring. In 2020, this result was extended by Nystedt and Öinert to nonassociative rings. On the other hand, in 2017, Lezama and Latorre introduced the concept of semi-graded rings, a family of algebraic structures that encompasses rings graded and besides has been a key tool for studying certain types of non-commutative rings. Since hypercentral groups generalize nilpotent groups, which in turn contain Abelian groups, we ask whether a result similar to that of Jespers holds in the semi-graded setting, i.e., whether the simplicity of such rings can be characterized in terms of a notion of semi-graded simplicity. This work aims to study, explore, and formulate questions and answers in this directioneng
dc.description.degreelevelMaestría
dc.description.degreenameMaestría en Ciencias - Matemáticas
dc.format.extentiv, 55 páginas
dc.format.mimetypeapplication/pdf
dc.identifier.instnameUniversidad Nacional de Colombiaspa
dc.identifier.reponameRepositorio Institucional Universidad Nacional de Colombiaspa
dc.identifier.repourlhttps://repositorio.unal.edu.co/spa
dc.identifier.urihttps://repositorio.unal.edu.co/handle/unal/88693
dc.language.isoeng
dc.publisherUniversidad Nacional de Colombia
dc.publisher.branchUniversidad Nacional de Colombia - Sede Bogotá
dc.publisher.facultyFacultad de Ciencias
dc.publisher.placeBogotá, Colombia
dc.publisher.programBogotá - Ciencias - Maestría en Ciencias - Matemáticas
dc.relation.referencesR. Abu-Dawwas. On Graded Semi-prime Rings. Proc. JangjeonMath. Soc., 20(1):19–22, 2017.
dc.relation.referencesV. Bavula. Description of bi-quadratic algebras on 3 generators with PBW basis. J. Algebra, 631:695–730, 2023.
dc.relation.referencesA. Bell. Localization and Ideal Theory in Noetherian Strongly Group-Graded Rings. J. Algebra, 105(1):76–115, 1987.
dc.relation.referencesA. Bell and K. Goodearl. Uniformrank over differential operator rings and Poincaré- Birkhoff-Witt extensions. Pacific J.Math., 131(1):13–37, 1988.
dc.relation.referencesK. Brown and K. R. Goodearl. Lectures on Algebraic Quantum Groups, volume 1 of Advanced Courses inMathematics - CRM Barcelona. Birkäuser-Verlag, 2002.
dc.relation.referencesJ. Bueso, J. Gómez-Torrecillas, and A. Verschoren. Algorithmic Methods in Non- Commutative Algebra: Applications to QuantumGroups. MathematicalModelling: Theory and Applications, Springer Dordrecht, 2003.
dc.relation.referencesV. Bardakov andM. Neshchadim. Upper central series for the group of unitriangular automorphisms of a free associative algebra. In C. S. Aravinda, W. M. Goldman, K. Gongopadhyay, A. Lubotzky, M. Mj, and A. Weaver., editors, Geometry, Groups and Dynamics. ICTS Program Groups, Geometry and Dynamics December 3–16, 2012 CEMS, Kumaun University, Almora, India, volume 639 of ContemporaryMathematics, pages 229–234. AmericanMathematical Society, 2015.
dc.relation.referencesA. Bell and S. Smith. Some 3-dimensional skew polynomial rings. University of Wisconsin,Milwaukee, preprint, 1990.
dc.relation.referencesM. Chebotar. On skew Laurent polynomial rings over locally nilpotent rings. Linear Algebra Appl., 549:287–290, 2018.
dc.relation.referencesP. M. Cohn. Quadratic extensions of skew fields. Proc. London Math. Soc., s3– 11(1):531–556, 1961.
dc.relation.referencesP. M. Cohn. Free Rings and Their Relations. Second Edition. Academic Press, London, 1985.
dc.relation.referencesS. C. Coutinho. A Primer of Algebraic D-modules. CambridgeUniversity Press. London Mathematical Society Student Texts, 1995.
dc.relation.referencesJ. H. Cozzens. Simple Principal Left Ideal Domains. J. Algebra, 23(1):66–75, 1972.
dc.relation.referencesA. Chacón and A. Reyes. Noncommutative scheme theory and the Serre-Artin-Zhang- Verevkin theorem for semi-graded rings. J. Noncommut. Geom., 19(2):495–532, 2025.
dc.relation.referencesE. Dade. Group-graded rings andmodules. Math Z., 174(1):241–262, 1980.
dc.relation.referencesD. Dummit and R. Foote. Abstract Algebra, volume 3. JohnWiley sons, Inc., 2004.
dc.relation.referencesR. Díaz and E. Pariguan. On the q-meromorphic Weyl algebra. São Paulo J. Math. Sci., 3(2):283–298, 2009.
dc.relation.referencesW. Fajardo, C. Gallego, O. Lezama, A. Reyes, H. Suárez, and H. Venegas. Skew PBW Extensions. Ring andModule-theoretic Properties,Matrix and Gröbner Methods, and Applications. Algebra and Applications. Springer, Cham, 2020.
dc.relation.referencesW. Fajardo, O. Lezama, C. Payares, A. Reyes, and C. Rodríguez. Introduction to Algebraic Analysis on Ore Extensions. In A.Martsinkovsky, editor, Functor Categories, Model Theory, Algebraic Analysis and ConstructiveMethods FCMTCCT2 2022, Almería, Spain, July 11-15, Invited and Selected Contributions, volume 450 of Springer Proceedings inMathematics & Statistics, pages 45–116. Springer, Cham, 2024.
dc.relation.referencesK. Goodearl and E. Letzter. Prime Ideals in Skew and q-Skew Polynomial Rings, volume 109 of Mem. Amer. Math. Soc. 521. Providence, Rhode Island: American Mathematical Society, Second edition, 1994.
dc.relation.referencesC. Gallego and O. Lezama. Gröbner Bases for Ideals of σ-PBW Extensions. Comm. Algebra, 39(1):50–75, 2011.
dc.relation.referencesK. Goodearl and R. Warfield. Primitivity in differential operator rings. Math Z., 80(4):503–527, 1982.
dc.relation.referencesK. Goodearl and R.Warfield. An Introduction to Noncommutative Noetherian Rings. Cambridge University Press. LondonMathematical Society Student Texts, 2004.
dc.relation.referencesA. Giaquinto and J. J. Zhang. Quantum Weyl Algebras. J. Algebra, 176(3):861–881, 1995.
dc.relation.referencesR.Hazrat. Graded Rings and Graded Grothendieck Groups. 435. LondonMathematical Society Lecture Note Series 435, 2016.
dc.relation.referencesK. Hrbacek and T. Jech. Introduction to Set Theory Groups. 3. Marcel Dekker, Inc., 2001.
dc.relation.referencesS. Higuera and A. Reyes. On weak annihilators and nilpotent associated primes of skew PBW extensions. Comm. Algebra, 51(11):4839–4861, 2023.
dc.relation.referencesS. Higuera and A. Reyes. Attached prime ideals over skew Ore polynomials. Comm. Algebra, 53(3):1076–1087, 2025.
dc.relation.referencesS. Higuera and A. Reyes. Uniformand couniformdimensions of inverse polynomial modules over skew Ore polynomials. Int. Electron. J. Algebra,2025.
dc.relation.referencesA. Isaev, P. Pyatov, and V. Rittenberg. Diffusion algebras. J. Phys. A., 34(29):5815–5834, 2001.
dc.relation.referencesN. Jacobson. Pseudo-linear transformations. Ann. ofMath., 38(2):484–507, 1937.
dc.relation.referencesA. Jategaonkar. A multiplicative analog of the Weyl algebra. Comm. Algebra., 12(14):1669–1688, 1984.
dc.relation.referencesA. Jannussis, G. Brodimas, and D. Sourlas. Remarks on the q-quantization. Lett. Nuovo Cimento, 30:123–127, 1981.
dc.relation.referencesE. Jespers. Simple graded rings. Comm. Algebra, 21(7):2437–2444, 1993.
dc.relation.referencesD. A. Jordan. Ore extensions and Jacobson rings. PhD thesis, University of Leeds, Leeds, England, 1975.
dc.relation.referencesD. A. Jordan. Primitive Ore extensions. Glasg.Math. J., 18(1):93–97, 1977.
dc.relation.referencesD. A. Jordan. Simple Skew Laurent Polynomial Rings. Comm. Algebra., 12(2):135–137, 1984.
dc.relation.referencesD. A. Jordan. Finite-dimensional simple modules over certain iterated skew polynomial rings. J. Pure Appl. Algebra, 98(1):45–55, 1995.
dc.relation.referencesD. A. Jordan. Down-Up Algebras and Ambiskew Polynomial Rings. J. Algebra, 228(1):311–346, 2000.
dc.relation.referencesD. A. Jordan and I. E.Wells. Invariants for automorphisms of certain iterated skew polynomial rings. Proc. Edinb.Math. Soc., 39(3):461–472, 1996.
dc.relation.referencesM. V. Kuryshkin. Opérateurs quantiques généralisés de création et d’annihilation. Ann. Fond. L. de Broglie, 5:111–125, 1980.
dc.relation.referencesO. Lezama. Computation of Point Modules of Finitely Semi-Graded Rings. Comm. Algebra, 48(2):866–878, 2020.
dc.relation.referencesO. Lezama. Cuadernos de Algebra, volume 1. Universidad Nacional de Colombia - Facultad de Ciencias, 2020.
dc.relation.referencesO. Lezama. Some Open Problems in the Context of Skew PBW Extensions and Semigraded Rings. Commun.Math. Stat., 9(1):347–378, 2021.
dc.relation.referencesO. Lezama and J. Gomez. Koszulity and PointModules of Finitely Semi-Graded Rings and Algebras. Symmetry, 1(1):1–22, 2019.
dc.relation.referencesH. Li. Noncommutative Gröbner Bases and Filtered-Graded Transfer. Lecture Notes in Mathematics. Springer Berlin, Heidelberg, 2002.
dc.relation.referencesO. Lezama and E. Latorre. Non-commutative algebraic geometry of semi-graded rings. Internat. J. Algebra Comput., 27(4):361–389, 2017.
dc.relation.referencesS. Lopes. Noncommutative Algebra and Representation Theory: Symmetry, Structure & Invariants. Commun.Math., 32(3):63–117, 2024.
dc.relation.referencesM. Louzari and A. Reyes. Generalized Rigid Modules and Their Polynomial Extensions. InM. SilesMolina, L. El Kaoutit, M. Louzari, L. Ben Yakoub, andM. Benslimane, editors, Associative and Non-Associative Algebras and Applications. MAMAA 2018, volume 311 of Springer Proceedings inMathematics & Statistics, pages 147–158. Springer, Cham, 2020.
dc.relation.referencesO. Lezama and H. Venegas. Center of skew PBW extensions. Internat. J. Algebra Comput., 30(08):1625–1650, 2020.
dc.relation.referencesT. Lee and Y. Zhou. Armendariz and reduced rings. Comm. Algebra, 32(6):2287–2299, 2004.
dc.relation.referencesY. I.Manin. Quantum Groups and Noncommutative Geometry.With a Contribution by Theo Raedschelders and Michel Van den Bergh. Centre de RecherchesMathématiques Short Courses. Springer Cham, Second edition, 2018.
dc.relation.referencesJ.McConnell and J. Robson. Noncommutative Noetherian Rings. Graduate Studies in Mathematics. Amer.Math. Soc., 2001.
dc.relation.referencesB. Nashier. Maximal Ideals in Laurent Polynomial Rings. Proc. Amer. Math. Soc., 115(4):907–913, 1992.
dc.relation.referencesP. Nystedt and J. Öinert. Simple graded rings, nonassociative crossed products and Cayley–Dickson doublings. J. Algebra Appl., 19(12):2050231, 2020.
dc.relation.referencesA. Niño and A. Reyes. On centralizers and pseudo-multidegree functions for noncommutative rings having PBW bases. J. Algebra Appl., 24(04):2550109, 2025.
dc.relation.referencesA. Niño,M. C. Ramírez, and A. Reyes. Associated prime ideals over skew PBW extensions. Comm. Algebra, 48(12):5038–5055, 2020.
dc.relation.referencesC. Nˇastˇasescu and F. van Oystaeyen. Methods of Graded Rings, volume 1. Springer Science BusinessMedia, 2004.
dc.relation.referencesJ. Öinert. Bimodules in group graded rings. Algebr. Represent. Theory, 20(6):483–1494, 2017.
dc.relation.referencesF. Oystaeyen and C. Nˇastˇasescu. Graded Ring Theory. North-Holland Pub. North- HollandMathematical Library, 1982.
dc.relation.referencesO. Ore. Linear Equations inNon-commutative Fields. Ann. ofMath. (2), 32(3):463–477, 1931.
dc.relation.referencesO. Ore. Theory of Non-Commutative Polynomials. Ann. ofMath. (2), 34(3):480–508, 1933.
dc.relation.referencesJ. Öinert, J. Richter, and S. Silvestrov. Maximal commutative subrings and simplicity of Ore extensions. J. Algebra Appl., 12(4):1250192, 2013.
dc.relation.referencesC. Phan. The Yoneda Algebra of a Graded Ore Extension. Comm. Algebra, 40:834–844, 2012.
dc.relation.referencesM. Refai and R. Abu-Dawwas. On properties of graded rings and graded modules. J.Math., 41:1413–1437, 2022.
dc.relation.referencesJ. Richter and P. Bäck. Simplicity of non-associative skew Laurent polynomial rings. Proc. Edinb.Math. Soc. (2), 68(1):1–15, 2025.
dc.relation.referencesJ. Richter. Hilbert’s Basis Theorem for Non-associative and Hom-associative Ore Extensions. Algebr. Represent. Theory, 26(1):1051–1065, 2023.
dc.relation.referencesD. Robinson. A Course in the Theory of Groups. Springer–Verlag, 1993.
dc.relation.referencesA. Rosenberg. Non-Commutative Algebraic Geometry and Representations of Quantized Algebras, volume 330 ofMathematics and Its Applications. Dordrecht, Springer, 1995.
dc.relation.referencesA. Reyes and C. Sarmiento. On the differential smoothness of 3-dimensional skew polynomial algebras and diffusion algebras. Internat. J. Algebra Comput., 32(3):529– 559, 2022.
dc.relation.referencesW.M. Seiler. Involution. The Formal Theory of Differential Equations and its Applications in Computer Algebra, volume 24 of Algorithms and Computation inMathematics (AACIM). Springer Berlin, Heidelberg, 2010.
dc.relation.referencesH. Smith. Hypercentral Groups with all Subgroups Subnormal. Lond. Math. Soc. Newsl., 15:229–234, 1983.
dc.relation.referencesS. P. Smith. Quantum Groups: An Introduction and Survey for Ring Theorists. In S.Montgometry and L. Small, editors, Noncommutative Rings, volume 24 ofMath. Sci. Res. Inst. Publ., pages 131–178. Sringer-Verlag, New York, 1992.
dc.relation.referencesJ.Wess and B. Zumino. Covariant differential calculus on the quantumhyperplane. Nucl. Phys. B Proc. Suppl., 18(2):302–312, 1991.
dc.rights.accessrightsinfo:eu-repo/semantics/openAccess
dc.rights.licenseReconocimiento 4.0 Internacional
dc.rights.urihttp://creativecommons.org/licenses/by/4.0/
dc.subject.bneÁlgebra abstractaspa
dc.subject.bneAlgebra, Abstracteng
dc.subject.ddc510 - Matemáticas::512 - Álgebra
dc.subject.lccAnillo no conmutativospa
dc.subject.lccNoncommutative ringseng
dc.subject.lccNilpotent groupseng
dc.subject.lccAnillos graduadosspa
dc.subject.lccGraded ringseng
dc.subject.lembAnillos (Álgebra)spa
dc.subject.lembRings (Algebra)spa
dc.subject.otherGrupos nilpotentesspa
dc.subject.proposalHypercentral groupeng
dc.subject.proposalGraded ringeng
dc.subject.proposalSemi-graded ringeng
dc.subject.proposalSimplicityeng
dc.subject.proposalGrupo hipercentralspa
dc.subject.proposalAnillo graduadospa
dc.subject.proposalAnillo semi-graduadospa
dc.subject.proposalSimplicidadspa
dc.subject.wikidataTeoría de anillosspa
dc.subject.wikidataRing theoryeng
dc.titleOn the simplicity of semi-graded rings by hypercentral groupseng
dc.title.translatedSobre la simplicidad de anillos semi-graduados por grupos hipercentralesspa
dc.typeTrabajo de grado - Maestría
dc.type.coarhttp://purl.org/coar/resource_type/c_bdcc
dc.type.coarversionhttp://purl.org/coar/version/c_ab4af688f83e57aa
dc.type.contentText
dc.type.driverinfo:eu-repo/semantics/masterThesis
dc.type.redcolhttp://purl.org/redcol/resource_type/TM
dc.type.versioninfo:eu-repo/semantics/acceptedVersion
oaire.accessrightshttp://purl.org/coar/access_right/c_abf2

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