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A Seventy Years Jubilee: The Hopkins-Levitzki Theorem

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Ring and Module Theory

Part of the book series: Trends in Mathematics ((TM))

Abstract

The aim of this expository paper is to discuss various aspects of the Hopkins-Levitzki Theorem (H-LT), including the Relative H-LT, the Absolute or Categorical H-LT, the Latticial H-LT, as well as the Krull dimension-like H-LT.

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References

  1. T. Albu, Sur la dimension de Gabriel des modules, Algebra — Berichte, Bericht Nr. 21, 1974, Seminar F. Kasch — B. Pareigis, Mathematisches Institut der Universität München, Verlag Uni-Druck, 26 pages.

    Google Scholar 

  2. T. Albu, Une remarque sur les catégories de Grothendieck commutatives, Bull. Math. Soc. Sci. Math. R. S. Roumanie 23 (71) (1979), 115–116.

    MathSciNet  Google Scholar 

  3. T. Albu, On commutative Grothendieck categories having a Noetherian cogenerator, Arch. Math. (Basel) 34 (1980), 210–219.

    MATH  MathSciNet  Google Scholar 

  4. T. Albu, Certain Artinian lattices are Noetherian. Applications to the relative Hopkins-Levitzki Theorem, in “Methods in Ring Theory”, edited by F. Van Oystaeyen, D. Reidel Publishing Company, Dordrecht (Holland), 1984, pp. 37–52.

    Google Scholar 

  5. T. Albu, Classes of lattices (co)generated by a lattice and their global (dual) Krull dimension, Discrete Math. 185 (1998), 1–18.

    Article  MATH  MathSciNet  Google Scholar 

  6. T. Albu, G. Krause, and M.L. Teply, The nilpotence of the τ-closed prime radical in rings with τ-Krull dimension, J. Algebra 229 (2000), 498–513.

    Article  MATH  MathSciNet  Google Scholar 

  7. T. Albu, T.H. Lenagan, and P.F. Smith, Ascending chains of submodules in modules having Krull dimension, Rev. Roumaine Math. Pures Appl. 41 (1996), 567–581.

    MATH  MathSciNet  Google Scholar 

  8. T. Albu and C. Năstăsescu, Décompositions primaires dans les catégories de Grothendieck commutatives (I), J. Reine Angew. Math. 280 (1976), 172–194.

    MATH  MathSciNet  Google Scholar 

  9. T. Albu and C. Năstăsescu, Déecompositions primaires dans les catégories de Grothendieck commutatives (II), J. Reine Angew. Math. 282 (1976), 172–185.

    MATH  MathSciNet  Google Scholar 

  10. T. Albu and C. Năstăsescu, “Relative Finiteness in Module Theory”, Marcel Dekker, Inc., New York and Basel, 1984.

    MATH  Google Scholar 

  11. T. Albu and P.F. Smith, Dual relative Krull dimension of modules over commutative rings, in “Abelian Groups and Modules”, edited by A. Facchini and C. Menini, Kluwer Academic Publisher, Dordrecht (Holland), 1995, pp. 1–15.

    Google Scholar 

  12. T. Albu and P.F. Smith, Localization of modular lattices, Krull dimension, and the Hopkins-Levitzki Theorem (I), Math. Proc. Cambridge Philos. Soc. 120 (1996), 87–101.

    Article  MATH  MathSciNet  Google Scholar 

  13. T. Albu and P.F. Smith, Localization of modular lattices, Krull dimension, and the Hopkins-Levitzki Theorem (II), Comm. Algebra 25 (1997), 1111–1128.

    Article  MATH  MathSciNet  Google Scholar 

  14. T. Albu and P.F. Smith, Dual Krull dimension and duality, Rocky Mountain J. Math. 29 (1999), 1153–1165.

    Article  MATH  MathSciNet  Google Scholar 

  15. T. Albu and P. Vámos, Global Krull dimension and global dual Krull dimension of valuation rings, in “Abelian Groups, Module Theory, and Topology: Proceedings in Honor of Adalberto Orsatti’s 60th Birthday”, edited by D. Dikranjan and L. Salce, Marcel Dekker, Inc., New York, 1998, pp. 37–54.

    Google Scholar 

  16. T. Albu and R. Wisbauer, Generators in Grothendieck categories with right perfect endomorphism rings, Osaka J. Math. 28 (1991), 295–304.

    MATH  MathSciNet  Google Scholar 

  17. E. Artin, Zur Theorie der hypercomplexen Zahlen, Abh. Math. Sem. Univ. Hamburg 5 (1927), 251–260.

    Article  MATH  Google Scholar 

  18. P. Crawley and R.P. Dilworth, “dAlgebraic Theory of Lattices”, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1973.

    Google Scholar 

  19. S.E. Dickson, A torsion theory for Abelian categories, Trans. Amer. Math. Soc. 121 (1966), 223–235.

    Article  MATH  MathSciNet  Google Scholar 

  20. C. Faith, “Injective Modules and Injective Quotient Rings”, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, Inc., New York and Basel, 1982.

    MATH  Google Scholar 

  21. L. Fuchs, Wann folgt die Maximalbedingung aus der Minimalbedingung?, Arch. Math. (Basel) 8 (1957), 317–319.

    MATH  MathSciNet  Google Scholar 

  22. P. Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448.

    MATH  MathSciNet  Google Scholar 

  23. J.S. Golan, “Torsion Theories”, Pitman/Longman, New York, 1986.

    Google Scholar 

  24. R. Gordon and J.C. Robson, “Krull Dimension”, Memoirs AMS, 133, 1973.

    Google Scholar 

  25. R. Gordon and J.C. Robson, The Gabriel dimension of a module, J. Algebra 29 (1974), 459–473.

    Article  MATH  MathSciNet  Google Scholar 

  26. G. Grätzer, “General Lattice Theory”, Second Edition, Birkhäuser Verlag, Basel Boston Berlin, 2003.

    MATH  Google Scholar 

  27. C. Hopkins, Rings with minimal condition for left ideals, Ann. Math. 40 (1939), 712–730.

    Article  MathSciNet  Google Scholar 

  28. B. Lemonnier, Déviation des ensembles et groupes abéliens totalement ordonnés, Bull. Sci. Math. 96 (1972), 289–303.

    MATH  MathSciNet  Google Scholar 

  29. B. Lemonnier, Dimension de Krull et codéviation. Application au théorème d’Eakin, Comm. Algebra 16 (1978), 1647–1665.

    Article  MathSciNet  Google Scholar 

  30. T.H. Lenagan, The nil radical of a ring with Krull dimension, Bull. London Math. Soc 5 (1973), 307–311.

    Article  MATH  MathSciNet  Google Scholar 

  31. J. Levitzki, On rings which satisfy the minimum condition for the right-hand ideals, Compositio Math. 7 (1939), 214–222.

    MATH  MathSciNet  Google Scholar 

  32. J.C. McConnell and J.C. Robson, “Noncommutative Noetherian Rings”, John Wiley & Sons, Chichester New York Brisbane Toronto Singapore, 1987.

    MATH  Google Scholar 

  33. C. Menini, On a question concerning locally Artinian categories, Comm. Algebra 17 (1987), 1357–1363.

    Article  MathSciNet  Google Scholar 

  34. C. Menini and A. Orsatti, The basic ring of a locally Artinian category and applications, Arch. Math. (Basel) 49 (1987), 484–496.

    MATH  MathSciNet  Google Scholar 

  35. R.W. Miller and M.L. Teply, The descending chain condition relative to a torsion theory, Pacific J. Math. 83 (1979), 207–220.

    MATH  MathSciNet  Google Scholar 

  36. T. Molien F.E. Molin, Über Systeme höherer komplexer Zahlen, Math. Ann. 41 (1893), 83–156.

    Article  Google Scholar 

  37. I. Murase, A condition for Artinian rings to be Noetherian, Canadian J. Math. 30 1978, 830–837.

    Article  MATH  MathSciNet  Google Scholar 

  38. C. Năstăsescu, Conditions de finitude pour les modules, Rev. Roumaine Math. Pures Appl. 24 (1979), 745–758.

    MATH  MathSciNet  Google Scholar 

  39. C. Năstăsescu, Théorème de Hopkins pour les catégories de Grothendieck, in “Ring Theory”, Proceedings of the 1980 Antwerp Conference, Lecture Notes in Mathematics 825, Springer-Verlag, Berlin Heidelberg New York, 1980, pp. 88–93.

    Google Scholar 

  40. C. Năstăsescu, Conditions de finitude pour les modules II, Rev. Roumaine Math. Pures Appl. 25 (1980), 615–630.

    MATH  MathSciNet  Google Scholar 

  41. C. Năstăsescu, Δ-Anneaux et modules Σ-injectifs. Applications aux catégories localement artinienns, Comm. Algebra 9 (1981), 1981–1996.

    Article  MATH  MathSciNet  Google Scholar 

  42. E. Noether, Idealtheorie in Ringbereichen, Math. Ann. 83 (1921), 24–66.

    Article  MATH  MathSciNet  Google Scholar 

  43. E. Noether, Hyperkomplexe Grössen und Darstellungstheorie, Math. Z. 30 (1929), 641–692.

    Article  MATH  MathSciNet  Google Scholar 

  44. M. Pouzet and N. Zaguia, Dimension de Krull des ensembles ordonnées, Discrete Math. 53 (1985), 173–192.

    Article  MATH  MathSciNet  Google Scholar 

  45. J.E. Roos, Locally Noetherian categories and generalized strictly linearly compact rings, in “Category Theory, Homology and Their Applications II”, Lecture Notes in Mathematics 92, Springer-Verlag, Berlin Heidelberg New York, 1969, pp. 197–277.

    Chapter  Google Scholar 

  46. R.C. Shock, Certain Artinian rings are Noetherian, Canadian J. Math. 24 1972, 553–556.

    Article  MATH  MathSciNet  Google Scholar 

  47. B. Stenström, “Rings of Quotients”, Springer-Verlag, Berlin Heidelberg New York, 1975.

    Google Scholar 

  48. H. Tominaga and I. Murase, A study on Artinian rings, Math. J. Okayama Univ. 21 (1979), 115–123.

    MATH  MathSciNet  Google Scholar 

  49. J.H.M. Wedderburn, On hypercomplex numbers, Proc. London Math. Soc. 6 (1908), 77–118.

    Article  Google Scholar 

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Dedicated to the memory of Mark L. Teply (1942 2006)

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Albu, T. (2010). A Seventy Years Jubilee: The Hopkins-Levitzki Theorem. In: Albu, T., Birkenmeier, G.F., Erdoğgan, A., Tercan, A. (eds) Ring and Module Theory. Trends in Mathematics. Springer, Basel. https://doi.org/10.1007/978-3-0346-0007-1_1

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