Direct-sum behavior of modules over one-dimensional rings

  • Ryan Karr
  • Roger Wiegand


Let R be a reduced, one-dimensional Noetherian local ring whose integral closure \(\overline{R}\) is finitely generated over R. Since \(\overline{R}\) is a direct product of finitely many principal ideal domains (one for each minimal prime ideal of R), the indecomposable finitely generated \(\overline{R}\)-modules are easily described, and every finitely generated \(\overline{R}\)-module is uniquely a direct sum of indecomposable modules. In this article we will see how little of this good behavior trickles down to R. Indeed, there are relatively few situations where one can describe all of the indecomposable R-modules, or even the torsion-free ones. Moreover, a given finitely generated module can have many different representations as a direct sum of indecomposable modules.


Local Ring Noetherian Ring Integral Closure Constant Rank Indecomposable Module 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Baeth, N.: A Krull-Schmidt theorem for one-dimensional rings with finite Cohen–Macaulay type. J. Pure Appl. Algebra 208, 923–940 (2007)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Baeth, N., Luckas, M.: Bounds for indecomposable torsion-free modules. preprintGoogle Scholar
  3. 3.
    Bass, H.: On the ubiquity of Gorenstein rings. Math. Z. 82, 8–28 (1963)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bruns, W., Herzog, J.: Cohen–Macaulay Rings, Cambridge Stud. in Adv. Math., vol.39. Cambridge University Press, Cambridge (1993)MATHGoogle Scholar
  5. 5.
    Buchweitz, R.-O., Greuel, G.-M., Schreyer, F.-O.: Cohen–Macaulay modules on hypersurface singularities II. Invent. Math. 88, 165–182 (1987)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cimen, N.: One-dimensional rings of finite Cohen–Macaulay type. Ph.D. Thesis, University of Nebraska (1994)Google Scholar
  7. 7.
    Cimen, N., Wiegand, R., Wiegand, S.: One-dimensional local rings of finite representation type. In: Facchini, A., Menini, C. (eds.) Abelian groups and modules. Kluwer (1995)Google Scholar
  8. 8.
    Crabbe, A., Saccon, S.: Manuscript in preparationGoogle Scholar
  9. 9.
    Dickson, L.E.: Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors. Am. J. Math. 35, 413–422 (1913)CrossRefGoogle Scholar
  10. 10.
    Dieterich, E.: Representation types of group rings over complete discrete valuation rings. In: Roggenkamp, K. (ed.) Integral representations and applications, Lecture Notes in Math. vol. 882. Springer, New York (1980)Google Scholar
  11. 11.
    Dieudonné, J., Grothendieck, A.: Éléments de Géométrie Algébrique IV, Partie 2. Publ.Math.I.H.E.S. 24 (1967)Google Scholar
  12. 12.
    Drozd, Ju.A., Roĭter, A.V.: Commutative rings with a finite number of indecomposable integral representations, (in Russian). Izv. Akad. Nauk. SSSR, Ser. Mat. 31, 783–798 (1967)MATHMathSciNetGoogle Scholar
  13. 13.
    Eisenbud, D., Evans, E.G. Jr.: Generating modules efficiently: theorems from algebraic K-theory. J. Algebra 27, 278–305 (1973)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Evans, E.G. Jr.: Krull-Schmidt and cancellation over local rings. Pacific J. Math. 46, 115–121 (1973)MATHMathSciNetGoogle Scholar
  15. 15.
    Facchini, A., Hassler, W., Klingler, L., Wiegand, R.: Direct-sum decompositions over one-dimensional Cohen–Macaulay local rings. In: Brewer, J., Glaz, S., Heinzer, W. (eds.) Multiplicative Ideal Theory in Commutative Algebra: A tribute to the work of Robert Gilmer. Springer (2006)Google Scholar
  16. 16.
    Green, E., Reiner, I.: Integral representations and diagrams. Michigan Math. J. 25, 53–84 (1978)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Greuel, G.-M., Knörrer, H.: Einfache Kurvensingularitäten und torsionfreie Moduln. Math. Ann. 270, 417–425 (1985)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Hassler, W.: Criteria for direct-sum cancellation, with an application to negative quadratic orders. J. Algebra 281, 395–405 (2004)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Hassler, W.: Direct-sum cancellation for modules over real quadratic orders. J. Pure Appl. Algebra 208, 575–589 (2007)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Hassler, W., Karr, R., Klingler, L., Wiegand, R.: Large indecomposable modules over local rings. J. Algebra 303, 202–215 (2006)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Hassler, W., Karr, R., Klingler, L., Wiegand, R.: Big indecomposable modules and direct-sum relations. Ill. J. Math. 51, 99–122 (2007)MATHMathSciNetGoogle Scholar
  22. 22.
    Hassler, W., Karr, R., Klingler, L., Wiegand, R.: Indecomposable modules of large rank over Cohen–Macaulay local rings. Trans.Am.Math.Soc. 360, 1391–1406 (2008)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Hassler, W., Wiegand, R.: Direct-sum cancellation for modules over one-dimensional rings. J. Algebra 283, 93–124 (2005)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Hassler, W., Wiegand, R.: Extended modules. J.Commut. Algebra (to appear). Available at
  25. 25.
    Karr, R.: Failure of cancellation for quartic and higher-degree orders. J. Algebra Appl. 1, 469–481 (2002)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Karr, R.: Finite representation type and direct-sum cancellation. J. Algebra 273, 734–752 (2004)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Kattchee, K.: Monoids and direct-sum decompositions over local rings. J.Algebra 256, 51–65 (2002)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Kronecker, L.: Über die congruenten Transformationen der bilinearen Formen. In: Hensel, K. (ed.) Monatsberichte Königl.Preuß.Akad.Wiss.Berlin, pp. 397–447 (1874) [reprinted in: Leopold Kroneckers Werke, vol. 1, pp. 423–483. Chelsea, New York (1968)]Google Scholar
  29. 29.
    Klingler, L., Levy, L.S.: Representation type of commutative Noetherian rings I: Local wildness. Pacific J.Math. 200, 345–386 (2001)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Klingler, L., Levy, L.S.: Representation type of commutative Noetherian rings II: Local tameness. Pacific J.Math. 200, 387–483 (2001)MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Klingler, L., Levy, L.S.: Representation type of commutative Noetherian rings III: Global wildness and tameness. Mem.Am.Math.Soc. (to appear)Google Scholar
  32. 32.
    Lech, C.: A method for constructing bad Noetherian local rings. Algebra, Algebraic Topology and their Interactions (Stockholm, 1983), Lecture Notes in Math. vol. 1183. Springer, Berlin (1986)Google Scholar
  33. 33.
    Leuschke, G., Wiegand, R.: Ascent of finite Cohen–Macaulay type. J. Algebra 228, 674–681 (2000)MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Leuschke, G., Wiegand, R.: Local rings of bounded Cohen–Macaulay type. Algebr. Represent. Theory 8, 225–238 (2005)MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Levy, L.S., Odenthal, C.J.: Package deal theorems and splitting orders in dimension 1. Trans.Am.Math.Soc. 348, 3457–3503 (1996)MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Reiner, I., Roggenkamp, K.W.: Integral Representations, Lecture Notes in Math., vol. 744. Springer, Berlin (1979)MATHGoogle Scholar
  37. 37.
    Serre, J.-P.: Faisceaux algébriques cohérents. Ann. Math. 61(2), 197–278 (1955) in FrenchCrossRefMathSciNetGoogle Scholar
  38. 38.
    Vasconcelos, W.V.: On local and stable cancellation. An.Acad.Brasil.Ci. 37, 389–393 (1965)MathSciNetGoogle Scholar
  39. 39.
    Warfield, R.B. Jr.: Decomposability of finitely presented modules. Proc.Am.Math.Soc. 25, 167–172 (1970)MATHMathSciNetGoogle Scholar
  40. 40.
    Weierstrass, K.: Zur Theorie der bilinearen und quadratischen Formen. Monatsberichte Königl.Preuß.Akad.Wiss.Berlin 310–338 (1968)Google Scholar
  41. 41.
    Wiegand, R.: Cancellation over commutative rings of dimension one and two. J. Algebra 88, 438–459 (1984)MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Wiegand, R.: Direct sum cancellation over commutative rings. Proc. Udine Conference on Abelian Groups and Modules, CISM 287, 241–266 (1985)Google Scholar
  43. 43.
    Wiegand, R.: Noetherian rings of bounded representation type, Commutative Algebra, Proceedings of a Microprogram (June 15 – July 2, 1987), pp. 497–516. Springer, New York (1989)Google Scholar
  44. 44.
    Wiegand, R.: Picard groups of singular affine curves over a perfect field. Math. Z. 200, 301–311 (1989)MATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Wiegand, R.: One-dimensional local rings with finite Cohen–Macaulay type, Algebraic Geometry and its Applications, pp. 381–389. Springer, New York (1994)Google Scholar
  46. 46.
    Wiegand, R.: Local rings of finite Cohen–Macaulay type. J.Algebra 203, 156–168 (1998)MATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    Wiegand, R.: Direct-sum decompositions over local rings. J.Algebra 240, 83–97 (2001)MATHCrossRefMathSciNetGoogle Scholar
  48. 48.
    Wiegand, R., Wiegand, S.: Stableisomorphismofmodulesoverone-dimensional rings. J. Algebra 107, 425–435 (1987)MATHCrossRefMathSciNetGoogle Scholar
  49. 49.
    Wiegand, R., Wiegand, S.: Semigroups of modules: a survey, Proceedings of the International Conference on Rings and Things, Contemp. Math.. Am. Math. Soc. (to appear). Available at

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Central FloridaOrlandoUSA
  2. 2.Department of MathematicsUniversity of NebraskaLincolnUSA

Personalised recommendations