Skip to main content
SpringerLink
Log in

Purity, Algebraic Compactness, Direct Sum Decompositions, and Representation Type

  • Conference paper
Infinite Length Modules

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

Abstract

The idea of purity pervades human culture, from religion through sexual and moral codes, cleansing rituals and dietary rules, to stratifications of societies into castes. It does so to an extent which is hardly backed up by the following definition found in the Oxford English Dictionary: freedom from admixture of any foreign substance or matter. In fact, in a process of blending and fusing of ideas and ideals (which stands in stark contrast to the dictionary meaning of the term itself), the concept has acquired dozens of additional connotations. They have proved notoriously hard to pin down, to judge by the incongruence of the emotional and poetic reactions they have elicited. Here is a small sample:

  • Purity is obscurity.Ogden Nash

  • One cannot be precise and still be pure. Marc Chagall.

  • Unto the pure all things are pure. New Testament

  • To the pure all things are impure. Mark Twain

  • To the pure all things are indecent. Oscar Wilde

  • ;Blessed are the pure in heart for they have so much more to talk about. Edith Wharton

  • Be thou as chaste as ice, as pure as snow, thou shalt not escape calumny. Get thee to a nunnery, go. W. Shakespeare.

  • Necessary, for ever necessary, to burn out false shames and smelt the heaviest ore of the body into purity. D. H. Lawrence.

  • Necessary, for ever necessary, to burn out false shames and smelt the heaviest ore of the body into purity.D. H. Lawrence.

  • Mathematics possesses not only truth, but supreme beauty - a beauty cold and austere, like that of a sculpture, without appeal to any part of our weaker nature, sublimely pure, and capable of a stern perfection such as only the greatest art can show. Bertrand Russell.

  • Purity strikes me as the most mysterious of the virtues and the more I think about it the less I know about it.Flannery O’Connor.

Helmut Lenzing zu seinem 60. Geburtstag gewidmet

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. F.W. Anderson and K.R. Fuller, Rings and Categories of Modules, Graduate Texts in Math. 13, Springer-Verlag, New York-Heidelberg, 1974; Second Edition, Springer-Verlag, New York, 1992.

    Google Scholar 

  2. M. Auslander, Representation theory of Artin Algebras II, Communic. in Algebra 1 (1974), 293–310.

    MathSciNet  Google Scholar 

  3. M. Auslander, Large modules over Artin algebras, in Algebra, topology and categories, Academic Press, New York, 1976, pp. 1–17.

    Google Scholar 

  4. M.Auslander and S.O. Sma1ø, Preprojective modules over Artin algebras, J. Algebra 66 (1980), 61–122.

    Article  MATH  MathSciNet  Google Scholar 

  5. G. Azumaya, Countable generatedness version of rings of pure global dimension zero, in: London Math. Soc. Lecture Notes Series, vol. 168, Cambridge University Press, 1992, pp. 43–79.

    MathSciNet  Google Scholar 

  6. G. Azumaya and A. Facchini, Rings of pure global dimension zero and Mittag-Leffler modules, J. Pure Appl. Algebra 62 (1989), 109–122.

    Article  MATH  MathSciNet  Google Scholar 

  7. D. Baer, H. Brune and H. Lenzing, A homological approach to representations of algebraas II: tame hereditary algebras, J. Pure and Appl. Algebra 26 (1982), 141–153.

    Article  MATH  MathSciNet  Google Scholar 

  8. D. Baer and H. Lenzing, A homological approach to representations of algebras I: the wild case, J. Pure and Appl. Algebra 24 (1982), 227–233.

    Article  MATH  MathSciNet  Google Scholar 

  9. S. Balcerzyk, On the algebraically compact groups of I. Kaplansky, Fund. Math. 44 (1957), 91–93.

    MATH  MathSciNet  Google Scholar 

  10. D.J. Benson, Infinite dimensional modules for finite groups, this volume.

    Google Scholar 

  11. D.J. Benson, M.C.R. Butler and G. Horrocks, Classes of extensions and resolutions, Philos. Trans. Roy. Soc., London 264 (1961), 155–222.

    Google Scholar 

  12. S.U. Chase, Direct products of modules, Trans. Amer. Math. Soc. 97 (1960), 457–473.

    Article  MathSciNet  Google Scholar 

  13. I.S. Cohen and I. Kaplansky, Rings for which every module is a direct sum of cyclic modules, Math. Zeitschr. 54 (1951), 97–101.

    Article  MATH  MathSciNet  Google Scholar 

  14. P.M. Cohn, On the free product of associative rings, Math. Zeitschr. 71 (1959), 380–398.

    Article  MATH  Google Scholar 

  15. P. Crawley and B. Jónnson, Refinements for infinite direct decompositions of algebraic systems, Pacific J. Math. 14 (1964), 797–855.

    Article  MATH  MathSciNet  Google Scholar 

  16. W. W. Crawley-Boevey, Tame algebras and generic modules, Proc. London Math. Soc. 63 (1991), 241–264.

    Article  MATH  MathSciNet  Google Scholar 

  17. W. W. Crawley-BoeveyModules of finite length over their endomorphism ringin Representations of Alge-bras and Related Topics (S. Brenner and H. Tachikawa, Eds., eds.), London Math. Soc. Lec. Note Series 168, Cambridge Univ. Press, Cambridge, 1992, pp. 127–184.

    Google Scholar 

  18. D. Eisenbud and P. Griffith, The structure of serial rings, Pacific J. Math. 36 (1971), 109–121.

    Article  MATH  MathSciNet  Google Scholar 

  19. A. Facchini, Anelli di tipo di rappresentazione finito, di dimensione pura globale zero, e moduli di Mittag-Leffler, Rend. Sem. Mat. Fis. Milano 59 (1989), 65–80.

    Article  MATH  MathSciNet  Google Scholar 

  20. A. Facchini, Mittag-Leffler modules, reduced products and direct products, Rend. Sem. Mat. Univ. Padova 85 (1991), 119–132.

    MATH  MathSciNet  Google Scholar 

  21. C. Faith, Rings with ascending condition on annihilators, Nagoya Math. J. 27 (1966), 179–191.

    MATH  MathSciNet  Google Scholar 

  22. D. J. Fieldhouse, Aspects of purity, in Ring Theory, Proc. Conf. Univ. Oklahoma 1973, Lecture Notes in Pure and Applied Math. 7, Dekker, New York, 1974, pp. 185–196.

    Google Scholar 

  23. L. Fuchs, Algebraically compact modules over Noetherian rings, Indian J. Math. 9 (1967), 357–374.

    MATH  MathSciNet  Google Scholar 

  24. L. Fuchs, Infinite Abelian Groups, Academic Press, New York and London, 1970.

    MATH  Google Scholar 

  25. K.R. Fuller, On rings whose left modules are direct sums of finitely generated modules Proc. Amer. Math. Soc. 54 (1976), 39–44.

    Article  MATH  MathSciNet  Google Scholar 

  26. K.R. Fuller and I. Reiten, Note on rings of finite representation type and decompositions of modules, Proc. Amer. Math. Soc. 50 (1975), 92–94.

    Article  MATH  MathSciNet  Google Scholar 

  27. S. Garavaglia, Dimension and rank in the theory of modules, Preprint, 1979.

    Google Scholar 

  28. P.A. Griffith, On the decomposition of modules and generalized left uniserial rings, Math. Ann. 184 (1970), 300–308.

    Article  MathSciNet  Google Scholar 

  29. L. Gruson, Simple coherent functors, in Representations of Algebras, Lecture Notes in Math. 488, Springer-Verlag, 1975, pp. 156–159.

    Google Scholar 

  30. L. Gruson and C.U. Jensen, Modules algébriquement compacts et foncteurs \({{\mathop{{\lim }}\limits_{ \leftarrow } }^{{\left( i \right)}}} \),C. R. Acad. Sci. Paris, Sér. A 276 (1973), 1651–1653.

    MATH  MathSciNet  Google Scholar 

  31. L. Gruson and C.U. Jensen, Deux applications de la notion de L-dimension, C. R. Acad. Sci. Paris, Sér. A 282(1976), 23–24.

    MATH  MathSciNet  Google Scholar 

  32. L. Gruson and C.U. Jensen Dimensions cohomologiques reliees aux foncteurs \({\mathop {\lim }\limits_ \leftarrow ^{(i)}} \), in Sém. d’Algèbre P. Dubreil et M.-P. Malliavin, Lecture Notes in Math. 867, Springer-Verlag, Berlin, 1981, pp. 234–249.

    Google Scholar 

  33. I. Herzog, Elementary duality for modules, Trans. Amer. Math. Soc. 340 (1993), 37–69.

    Article  MATH  MathSciNet  Google Scholar 

  34. I. Herzog, A test for finite representation type, J. Pure Appl. Algebra 95 (1994), 151–182.

    Article  MATH  MathSciNet  Google Scholar 

  35. B. Huisgen-Zimmermann, Rings whose right modules are direct sums of indecomposable modules, Proc. Amer. Math. Soc. 77 (1979), 191–197.

    Article  MathSciNet  Google Scholar 

  36. B. Huisgen-Zimmermann, Strong preinjective partitions and representation type of artinian rings, Proc. Amer Math. Soc. 109 (1990), 309–322.

    Article  MathSciNet  Google Scholar 

  37. B. Huisgen-Zimmermann and W. Zimmermann, Algebraically compact rings and modules, Math. Zeitschr. 161 (1978), 81–93.

    Article  Google Scholar 

  38. B. Huisgen-Zimmermann and W. ZimmermannClasses of modules with the exchange property J. lgebra 88 (1984), 416–434.

    Article  MathSciNet  Google Scholar 

  39. B. Huisgen-Zimmermann and W. Zimmermann, On the sparsity of representations of rings of pure global dimension zero, Trans. Amer. Math. Soc. 320 (1990), 695–711.

    Article  MathSciNet  Google Scholar 

  40. B. Huisgen-Zimmermann and W. Zimmermann, On the abundance of ℵ 1 -separable modules, in Abelian Groups and Noncommutative Rings, A Collection of Papers in Memory of Robert B. Warfield, Jr. (L. Fuchs, K.R. odearl, J.T. Stafford, and C. Vinsonhaler, Eds.), Contemp. Math. 130 (1992), 167–180.

    Chapter  Google Scholar 

  41. H. Hullinger, Stable equivalence and rings whose modules are a direct sum of finitely generated modules, J. Pure Appl. Algebra 16 (1980), 265–273.

    Article  MATH  MathSciNet  Google Scholar 

  42. C.U. Jensen and H. LenzingAlgebraic compactness of reduced products and applications to pure global dimensionComm. Algebra11(1983), 305–325.

    Article  MATH  MathSciNet  Google Scholar 

  43. C.U. Jensen and H. LenzingModel theoretic algebra with particular emphasis on fields, rings, modulesAlgebra,Logic and Applications, vol. 2, Gordon & Breach Science Publishers, 1989

    Google Scholar 

  44. C.U. Jensen and B. Zimmermann-HuisgenAlgebraic compactness of ultrapowers and representation typePac. J. Math. 139 (1989), 251–265

    Article  MATH  MathSciNet  Google Scholar 

  45. I. Kielpiński, Infinite Abelian Groups, Univ. of Michigan Press, Ann Arbor, 1954.

    Google Scholar 

  46. R. Kielpiński, On Γ-pure injective modules, Bulletin de L’Academie Polonaise des Sciences, Sér. des sciences math., astr. et phys. 15 (1967), 127–131.

    MATH  Google Scholar 

  47. R. Kielpiński and D. Simson, On pure homological dimension, Bulletin de L’Académie Polonaise des Sciences, Sér. des sciences math., astr. et phys. 23 (1975), 1–6.

    MATH  Google Scholar 

  48. G. KoetheVerallgemeinerte abelsche Gruppen mit hyperkomplexem Operatorenring Math. Zeitschr. 39(1935), 31–44.

    Article  Google Scholar 

  49. H. KrauseGeneric modules over Artin algebras Proc. London Math. Soc. 76 (1998), 276–306.

    Article  Google Scholar 

  50. H. Krause, Finite versus infinite dimensional representations — a new definition of tameness,this volume.

    Google Scholar 

  51. H. Krause and M. Saorín, On minimal approximations of modules, in Trends in the Representation Theory of Finite Dimensional Algebras (E.L. Green and B. Huisgen-Zimmermann, Eds.), Contemp. Math. 229 (1998), 227–236.

    Google Scholar 

  52. A. Laradji, On duo rings, pure semisimplicity and finite representation type, Communic. in Algebra 25 (1997), 3947–3952.

    Article  MATH  MathSciNet  Google Scholar 

  53. J. Loś, Abelian groups that are direct summands of every abelian group which contains them as pure subgroups, Fund. Math. 44 (1957), 84–90.

    MATH  MathSciNet  Google Scholar 

  54. J. M. Maranda, On pure subgroups of abelian groups, Arch. Math. 11 (1960), 1–13.

    Article  MATH  MathSciNet  Google Scholar 

  55. J. Mycielski, Some compactifications of general algebras, Colloq. Math. 13 (1964), 1–9.

    MATH  MathSciNet  Google Scholar 

  56. F. Okoh, Hereditary algebras that are not pure hereditary, in Representation theory II, Proc. 2nd ICRA, Ottawa 1979, Lecture Notes in Math. 832, Springer-Verlag, Berlin-New York, 1980, pp. 432–437.

    Google Scholar 

  57. B. L. Osofsky, Endomorphism rings of quasi-injective modules, Canad. J. Math. 20 (1968), 895–903.

    Article  MATH  MathSciNet  Google Scholar 

  58. M. Prest, Duality and pure semisimple rings, J. London Math. Soc. 38 (1988), 403–409.

    MATH  MathSciNet  Google Scholar 

  59. M. Prest, Topological and geometric aspects of the Ziegler spectrum, this volume.

    Google Scholar 

  60. C.M. Ringel, Representations of K-species and bimodules, J. Algebra 41 (1976), 269–302.

    Article  MATH  MathSciNet  Google Scholar 

  61. C.M. Ringel, A construction of endo finite modules, Preprint.

    Google Scholar 

  62. C.M. Ringel and H. Tachikawa, QF-3 rings, J. reine angew. Math. 272 (1975), 49–72.

    MATH  MathSciNet  Google Scholar 

  63. M. Schmidmeier, The local duality for homomorphisms and an application to pure semisimple PI-rings, Colloq. Math. 77 (1998), 121–132.

    MATH  MathSciNet  Google Scholar 

  64. R. Schulz, Reflexive modules over perfect ringsJ. Algebra61 (1979), 527–537.

    Article  Google Scholar 

  65. D. Simson, Pure semisimple categories and rings of finite representation type, J. Algebra 48 (1977), 290–296; Corrigendum, J. Algebra 67 (1980), 254–256.

    MATH  MathSciNet  Google Scholar 

  66. D. Simson, On pure global dimension of locally finitely presented Grothendieck categories, Fund. Math. 96 (1977), 91–116.

    MATH  MathSciNet  Google Scholar 

  67. D. SimsonPartial Coxeter functors and right pure semisimple hereditary rings, J. Algebra 71 (1981), 195–218.

    Article  MATH  MathSciNet  Google Scholar 

  68. D. Simson, On right pure semisimple hereditary rings and an Artin problem, J. Pure Appl. Algebra104 (1995), 313–332.

    Article  MATH  MathSciNet  Google Scholar 

  69. D. Simson, A class of potential counterexamples to the pure semisimplicity conjecture, in Ad-vances in Algebra and Model Theory (M. Droste and R. Gael, eds.), Algebra Logic and Applications Series 9, Gordon and Breach, Amsterdam, 1997, pp. 345–373.

    Google Scholar 

  70. D. Simson, Dualities and pure semisimple rings, in Abelian Groups, Module Theory, and Topol-ogy (Padova 1997) (D. Dikranjan and L. Salce, eds.), Lecture Notes in Pure and Appl. Math., vol. 201, Dekker, New York, 1998, pp. 381–388.

    Google Scholar 

  71. B. Stenström, Pure submodules, Arkiv for Mat. 7 (1967), 159–171.

    Article  MATH  Google Scholar 

  72. R.B. Warfield, Jr.Purity and algebraic compactness for modules Pac. J. Math. 28(1969), 699–719.

    Article  MATH  MathSciNet  Google Scholar 

  73. R.B. WarfieldExchange rings and decompositions of modulesMath. Annalen199 (1972), 31–36.

    Article  MATH  MathSciNet  Google Scholar 

  74. R.B. WarfieldRings whose modules have nice decompositions, Math. Zeitschr. 125 (1972), 187–192.

    Article  MATH  MathSciNet  Google Scholar 

  75. B. WeglorzEquationally compact algebras, I, Fund. Math. 59 (1966), 289–298.

    MATH  MathSciNet  Google Scholar 

  76. M. ZayedIndecom,posable modules over right pure semisimple rings, Monatsh. Math. 105 (1988), 165–170.

    MATH  MathSciNet  Google Scholar 

  77. D. Zelinsky, Linearly compact modules and rings, Amer. J. Math. 75 (1953), 79–90.

    Article  MATH  MathSciNet  Google Scholar 

  78. W. Zimmermann, Einige Charakterisierungen üder Ringe fiber denen reine Untermoduln direkte Summanden sind, Bayer. Akad. Wiss. Math.-Natur. Kl. S.-B. 1972, Abt. II (1973), 77–79.

    Google Scholar 

  79. W. Zimmermann, Rein-injektive direkte Summen von Moduln, Habilitationsschrift, Universitat Munchen,1975.

    Google Scholar 

  80. W. Zimmermann,Rein-injektive direkte Summen von ModulnCommunic. in Algebra 5 (1977), 1083–1117

    Article  MATH  Google Scholar 

  81. W. Zimmermann, (Σ-) algebraic compactness of rings, J. Pure Appl. Algebra 23 (1982), 319–328.

    Article  MATH  MathSciNet  Google Scholar 

  82. W. Zimmermann, Modules with chain conditions for finite matrix subgroups, J. Algebra 190, 68–87.

    Google Scholar 

  83. W. Zimmermann, Extensions of three classical theorems to modules with maximum condition for finite matrix subgroups, forum Math. 10 (1998), 377–392.

    Article  MATH  MathSciNet  Google Scholar 

  84. G. Zwara, Tame algebras and degenerations of modules, this volume.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, Santa Barbara, CA, 93106, USA

    Birge Huisgen-Zimmermann

Authors
  1. Birge Huisgen-Zimmermann
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Fakultät für Mathematik, Universität Bielefeld, D-33501, Bielefeld, Germany

    Henning Krause  & Claus Michael Ringel  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer Basel AG

About this paper

Cite this paper

Huisgen-Zimmermann, B. (2000). Purity, Algebraic Compactness, Direct Sum Decompositions, and Representation Type. In: Krause, H., Ringel, C.M. (eds) Infinite Length Modules. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8426-6_18

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8426-6_18

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9562-0

  • Online ISBN: 978-3-0348-8426-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation