Bulletin of Mathematical Sciences

, Volume 2, Issue 2, pp 225–279 | Cite as

Direct-sum decompositions of modules with semilocal endomorphism rings

  • Alberto Facchini
Open Access


According to the classical Krull–Schmidt Theorem, any module of finite composition length decomposes as a direct sum of indecomposable modules in an essentially unique way, that is, unique up to isomorphism of the indecomposable summands and a permutation of the summands. Modules that do not have finite composition length can have completely different behaviors. In this survey, we consider in particular the case of the modules M R whose endomorphism ring E := End(M R ) is a semilocal ring, that is, E/J(E) is a semisimple artinian ring. For instance, modules of finite composition length have a semilocal endomorphism ring, but several other classes of modules also have a semilocal endomorphism ring, for example artinian modules, finite direct sums of uniserial modules, finitely generated modules over commutative semilocal rings, and finitely presented modules over arbitrary semilocal rings. Several interesting phenomena appear in these cases. For instance, modules with a semilocal endomorphism ring have very regular direct-sum decompositions into indecomposables, their direct summands can be described via lattices, and direct-sum decompositions into indecomposables (=uniserial submodules) of finite direct sums of uniserial modules are described via their monogeny classes and their epigeny classes up to two permutations of the factors.


Direct-sum decomposition Module Semilocal ring Endomorphism ring Uniserial module Artinian module 

Mathematics Subject Classification

16D70 16D90 16L30 18E05 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.


  1. 1.
    Amini B., Amini A., Facchini A.: Equivalence of diagonal matrices over local rings. J. Algebra 320, 1288–1310 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Amini A., Amini B., Facchini A.: Cyclically presented modules over rings of finite type. Commun. Algebra 39, 76–99 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Amini A., Amini B., Facchini A.: Direct summands of direct sums of modules whose endomorphism rings have two maximal right ideals. J. Pure Appl. Algebra 215, 2209–2222 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Anderson F.W., Fuller K.R.: Rings and Categories of Modules, 2nd edn. Springer, New York (1992)zbMATHCrossRefGoogle Scholar
  5. 5.
    Márki P.N., Márki L.: Morita equivalence for rings without identity. Tsukuba J. Math. 11, 1–16 (1987)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Ara P., Facchini A.: Direct sum decompositions of modules, almost trace ideals, and pullbacks of monoids. Forum Math. 18, 365–389 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Arnold D., Hunter R., Richman F.: Global Azumaya theorems in additive categories. J. Pure Appl. Algebra 16, 223–242 (1980)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Azumaya G.: Corrections and supplementaries to my paper concerning Krull-Remak-Schmidt’s theorem. Nagoya Math. J. 1, 117–124 (1950)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Bass H.: Algebraic K-Theory. Benjamin, New York (1968)zbMATHGoogle Scholar
  10. 10.
    Bergman G.M.: Coproducts and some universal ring constructions. Trans. Am. Math. Soc. 200, 33–88 (1974)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Bergman G.M., Dicks W.: Universal derivations and universal ring constructions. Pac. J. Math. 79, 293–337 (1978)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bican L.: Weak Krull-Schmidt theorem. Comment. Math. Univ. Carolin. 39, 633–643 (1998)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Brookfield G.: A Krull-Schmidt theorem for Noetherian modules. J. Algebra 251, 70–79 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Bumby R.T.: Modules which are isomorphic to submodules of each other. Arch. Math. 16, 184–185 (1965)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Camps R., Dicks W.: On semilocal rings. Isr. J. Math. 81, 203–211 (1993)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Camps R., Menal P.: Power cancellation for artinian modules. Commun. Algebra 19, 2081–2095 (1991)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Chouinard L.G. II: Krull semigroups and divisor class groups. Can. J. Math. 33, 1459–1468 (1981)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Cohn P.M.: Skew Field Constructions. Cambridge University Press, Cambridge (1977)zbMATHGoogle Scholar
  19. 19.
    Dauns J., Fuchs L.: Infinite Goldie dimension. J. Algebra 115, 297–302 (1988)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Diracca L., Facchini A.: Uniqueness of monogeny classes for uniform objects in abelian categories. J. Pure Appl. Algebra 172, 183–191 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Dress A.: On the decomposition of modules. Bull. Am. Math. Soc. 75, 984–986 (1969)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Dung N.V., Facchini A.: Weak Krull-Schmidt for infinite direct sums of uniserial modules. J. Algebra 193, 102–121 (1977)CrossRefGoogle Scholar
  23. 23.
    Dung N.V., Facchini A.: Direct summands of serial modules. J. Pure Appl. Algebra 133, 93–106 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Facchini A.: Krull-Schmidt fails for serial modules. Trans. Am. Math. Soc. 348, 4561–4575 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Facchini, A.: Module Theory. Endomorphism Rings and Direct Sum Decompositions in Some Classes of Modules. Birkhäuser Verlag, Basel (1998)Google Scholar
  26. 26.
    Facchini, A.: A characterization of additive categories with the Krull-Schmidt property. In: Huynh, D.V., Jain, S., López-Permouth, S.R. (eds.) Algebra and its Applications, pp. 125–129. Contemporary Math. Series 419. Amer. Math. Soc., Providence (2006)Google Scholar
  27. 27.
    Facchini, A.: Krull monoids and their application in module theory. In: Facchini, A., Fuller, K., Ringel, C. M., Santa-Clara, C. (eds.) Algebras, Rings and Their representations, pp. 53–71. World Scientific, Singapore (2006)Google Scholar
  28. 28.
    Facchini A.: Representations of additive categories and direct-sum decompositions of objects. Indiana Univ. Math. J. 56, 659–680 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Facchini, A.: Injective modules, spectral categories, and applications. In: Jain, S. K., Parvathi, S. (eds.) Noncommutative Rings, Group Rings, Diagram Algebras and Their Applications, pp. 1–17. Contemporary Math. 456. Amer. Math. Soc., Providence (2008)Google Scholar
  30. 30.
    Facchini A., Ecevit Ş., Koşan M.T.: Kernels of morphisms between indecomposable injective modules. Glasgow Math. J. 52A, 69–82 (2010)zbMATHCrossRefGoogle Scholar
  31. 31.
    Facchini, A., Girardi, N.: Couniformly presented modules and dualities. In: Huynh, D.V., López Permouth, S.R. (eds.) Advances in Ring Theory, pp. 149–163. Trends in Math. Birkhäuser Verlag, Basel (2010)Google Scholar
  32. 32.
    Facchini, A., Girardi, N.: Auslander-Bridger modules. Commun. Algebra 40(7) (2012). doi: 10.1080/00927872.2011.579588
  33. 33.
    Facchini A., Halter-Koch F.: Projective modules and divisor homomorphisms. J. Algebra Appl. 2, 435–449 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Facchini, A., Hassler, W., Klingler, L., Wiegand, R.: Direct-sum decompositions over one-dimensional Cohen-Macaulay local rings. In: Brewer, J.W., Glaz, S., Heinzer, W.J., Olberding, B.M. (eds.) Multiplicative Ideal Theory in Commutative algebra, pp. 153–168. Springer, New York (2006)Google Scholar
  35. 35.
    Facchini A., Herbera D.: K0 of a semilocal ring. J. Algebra 225, 47–69 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Facchini, A., Herbera, D.: Projective modules over semilocal rings. In: Huynh, D.V., Jain, S.K., López-Permouth, S.R. (eds.) Algebra and its Applications, pp. 181–198. Contemporary Mathematics 259. Amer. Math. Soc., Providence (2000)Google Scholar
  37. 37.
    Facchini A., Herbera D.: Two results on modules whose endomorphism ring is semilocal. Algebr. Represent. Theory 7, 575–585 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    Facchini A., Herbera D.: Local morphisms and modules with a semilocal endomorphism ring. Algebr. Represent. Theory 9, 403–422 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    Facchini A., Herbera D., Levy L.S., Vámos P.: Krull-Schmidt fails for artinian modules. Proc. Am. Math. Soc. 123, 3587–3592 (1995)zbMATHCrossRefGoogle Scholar
  40. 40.
    Facchini A., Perone M.: Maximal ideals in preadditive categories and semilocal categories. J. Algebra Appl. 10, 1–27 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  41. 41.
    Facchini A., Příhoda P.: Factor categories and infinite direct sums. Int. Electron. J. Algebra 5, 135–168 (2009)zbMATHMathSciNetGoogle Scholar
  42. 42.
    Facchini A., Příhoda P.: Endomorphism rings with finitely many maximal right ideals. Commun. Algebra 39, 3317–3338 (2011)zbMATHCrossRefGoogle Scholar
  43. 43.
    Facchini A., Příhoda P.: The Krull-Schmidt Theorem in the case two. Algebr. Represent. Theory 14, 545–570 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    Facchini A., Wiegand R.: Direct-sum decompositions of modules with semilocal endomorphism ring. J. Algebra 274, 689–707 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  45. 45.
    Faith C., Herbera D.: Endomorphism rings and tensor products of linearly compact modules. Commun. Algebra 25, 1215–1255 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  46. 46.
    Feferman, S.: Enriched stratified systems for the foundations of category theory. In: Sica, G. (ed.) What is Category Theory?, pp. 185–203. Polimetrica, Milano (2006)Google Scholar
  47. 47.
    Frobenius G., Stickelberger L.: Über Gruppen von vertauschbaren Elementen. J. Reine Angew. Math. 86, 217–262 (1879)MathSciNetGoogle Scholar
  48. 48.
    Fuchs L., Salce L.: Uniserial modules over valuation rings. J. Algebra 85, 14–31 (1983)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Fuller K.R.: On rings whose left modules are direct sums of finitely generated modules. Proc. Am. Math. Soc. 54, 39–44 (1976)zbMATHMathSciNetCrossRefGoogle Scholar
  50. 50.
    Gabriel P.: Des catégories abéliennes. Bull. Soc. Math. France 90, 323–448 (1962)zbMATHMathSciNetGoogle Scholar
  51. 51.
    Gabriel P., Oberst U.: Spektralkategorien und reguläre Ringe im von-Neumannschen Sinn. Math. Z. 92, 389–395 (1966)zbMATHMathSciNetCrossRefGoogle Scholar
  52. 52.
    Goldie A.W.: The structure of prime rings under ascending chain conditions. Proc. Lond. Math. Soc. 8(3), 589–608 (1958)zbMATHMathSciNetCrossRefGoogle Scholar
  53. 53.
    Grätzer G.: General Lattice Theory. Academic Press, New York (1978)CrossRefGoogle Scholar
  54. 54.
    Grzeszczuk P., Puczyłowski E.R.: On Goldie and dual Goldie dimensions. J. Pure Appl. Algebra 31, 47–54 (1984)zbMATHMathSciNetCrossRefGoogle Scholar
  55. 55.
    Halter-Koch F.: Ideal Systems. An Introduction to Multiplicative Ideal Theory. Marcel Dekker, New York (1998)zbMATHGoogle Scholar
  56. 56.
    Hassler W., Karr R., Klingler L., Wiegand R.: Big indecomposable modules and direct-sum relations. Illinois J. Math. 51, 99–122 (2007)zbMATHMathSciNetGoogle Scholar
  57. 57.
    Herbera D., Příhoda P.: Big projective modules over noetherian semilocal rings. J. Reine Angew. Math. 648, 111–148 (2010)zbMATHMathSciNetGoogle Scholar
  58. 58.
    Herbera, D., Příhoda, P.: Infinitely generated projective modules over pullbacks of rings. arXiv:1105.3627v1 [math.RA]Google Scholar
  59. 59.
    Herbera D., Shamsuddin A.: Modules with semi-local endomorphism ring. Proc. Am. Math. Soc. 123, 3593–3600 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  60. 60.
    Krause U.: On monoids of finite real character. Proc. Am. Math. Soc. 105, 546–554 (1989)zbMATHCrossRefGoogle Scholar
  61. 61.
    Krull W.: Über verallgemeinerte endliche Abelsche Gruppen. Math. Z. 23, 161–196 (1925)zbMATHMathSciNetCrossRefGoogle Scholar
  62. 62.
    Krull W.: Matrizen, Moduln und verallgemeinerte Abelsche Gruppen im Bereich der ganzen algebraischen Zahlen. Heidelberger Akademie der Wissenschaften 2, 13–38 (1932)Google Scholar
  63. 63.
    Leuschke, G.J., Wiegand, R.: Cohen-Macaulay Representations. Math. Surveys and Monographs, vol. 181. Amer. Math. Soc., Providence (2012)Google Scholar
  64. 64.
    Menal, P.: Cancellation modules over regular rings. In: Bueso, J., Jara, P., Torrecillas, B. (eds.) Ring Theory (Granada, 1986), pp. 187–208. Lecture Notes in Math., vol. 1328. Springer, Berlin (1988)Google Scholar
  65. 65.
    Mitchell B.: Rings with several objects. Adv. Math. 8, 1–161 (1972)zbMATHCrossRefGoogle Scholar
  66. 66.
    Maclagan-Wedderburn J.H.: On the direct product in the theory of finite groups. Ann. Math. 10, 173–176 (1909)zbMATHMathSciNetCrossRefGoogle Scholar
  67. 67.
    Mac Lane S.: Categories for the Working Mathematician, 2nd edn. Springer, New York (1997)Google Scholar
  68. 68.
    Ore O.: Direct decompositions. Duke Math. J. 2, 581–596 (1936)MathSciNetCrossRefGoogle Scholar
  69. 69.
    Pimenov, K.I., Yakovlev, A.V.: Artinian modules over a matrix ring. In: Krause, H., Ringel, C.M. (eds.) Infinite Length Modules. Trends in Mathematics. Birkhäuser Verlag, Basel (2000)Google Scholar
  70. 70.
    Příhoda P.: Weak Krull-Schmidt theorem and direct sum decompositions of serial modules of finite Goldie dimension. J. Algebra 281, 332–341 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  71. 71.
    Příhoda P.: A version of the weak Krull-Schmidt theorem for infinite direct sums of uniserial modules. Commun. Algebra 34, 1479–1487 (2006)zbMATHCrossRefGoogle Scholar
  72. 72.
    Příhoda P.: Projective modules are determined by their radical factors. J. Pure Appl. Algebra 210, 827–835 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  73. 73.
    Příhoda P.: Fair-sized projective modules. Rend. Sem. Mat. Univ. Padova 123, 141–167 (2010)zbMATHCrossRefGoogle Scholar
  74. 74.
    Příhoda P., Puninski G.: Non-finitely generated projective modules over generalized Weyl algebras. J. Algebra 321, 1326–1342 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  75. 75.
    Puninski G.: Some model theory over an exceptional uniserial ring and decompositions of serial modules. J. Lond. Math. Soc. 64(2), 311–326 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  76. 76.
    Puninski G.: Some model theory over a nearly simple uniserial domain and decompositions of serial modules. J. Pure Appl. Algebra 163, 319–337 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  77. 77.
    Puninski G.: Projective modules over the endomorphism ring of a biuniform module. J. Pure Appl. Algebra 188, 227–246 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  78. 78.
    Ringel C.M.: Krull-Remak-Schmidt fails for Artinian modules over local rings. Algebr. Represent. Theory 4, 77–86 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  79. 79.
    Schmidt O.: Sur les produits directs. Bull. Soc. Math. France 41, 161–164 (1913)zbMATHMathSciNetGoogle Scholar
  80. 80.
    Schmidt O.: Über unendliche Gruppen mit endlicher Kette. Math. Z. 29, 34–41 (1929)MathSciNetCrossRefGoogle Scholar
  81. 81.
    Stenström B.: Rings of Quotients. Springer, New York (1975)zbMATHCrossRefGoogle Scholar
  82. 82.
    Walker C.L., Warfield R.B.: Unique decomposition and isomorphic refinement theorems in additive categories. J. Pure Appl. Algebra 7, 347–359 (1976)zbMATHMathSciNetCrossRefGoogle Scholar
  83. 83.
    Warfield R.B.: Decompositions of injective modules. Pac. J. Math. 31, 263–276 (1969)zbMATHMathSciNetCrossRefGoogle Scholar
  84. 84.
    Warfield R.B.: Serial rings and finitely presented modules. J. Algebra 37, 187–222 (1975)zbMATHMathSciNetCrossRefGoogle Scholar
  85. 85.
    Warfield R.B.: Cancellation of modules and groups and stable range of endomorphism rings. Pac. J. Math. 91, 457–485 (1980)zbMATHMathSciNetCrossRefGoogle Scholar
  86. 86.
    Wehrung F.: Non-measurability properties of interpolation vector spaces. Isr. J. Math. 103, 177–206 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  87. 87.
    Wiegand, R.: Singularities and direct-sum decompositions. In: Grant Melles, C., Michler, R.I. (eds.) Singularities in Algebraic and Analytic Geometry, pp. 29–43. Contemp. Math., vol. 266. Amer. Math. Soc., Providence (2000)Google Scholar
  88. 88.
    Wiegand R.: Direct-sum decompositions over local rings. J. Algebra 240, 83–97 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  89. 89.
    Wiegand, R., Wiegand, S.: Semigroups of modules: a survey. In: Dung, N.V., Guerriero, F., Hammoudi, L., Kanwar, P. (eds.) Rings, Modules and Representations, pp. 335–349. Contemp. Math., vol. 480. Amer. Math. Soc., Providence (2009)Google Scholar
  90. 90.
    Yakovlev, A.V.: On direct sum decompositions of Artinian modules. Algebra i Analiz 10, 229–238 (1998). Translation in St. Petersburg Math. J. 10, 399–406 (1999)Google Scholar

Copyright information

© The Author(s) 2012

Authors and Affiliations

  1. 1.Dipartimento di Matematica Pura e ApplicataUniversità di PadovaPaduaItaly
  2. 2.Middle East Center of Algebra and its Applications (MECAA)King Abdulaziz UniversityJeddahSaudi Arabia

Personalised recommendations