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Two examples of local Artinian rings


Author: Wei Min Xue
Journal: Proc. Amer. Math. Soc. 107 (1989), 63-65
MSC: Primary 16A35; Secondary 16A52
DOI: https://doi.org/10.1090/S0002-9939-1989-0979222-2
MathSciNet review: 979222
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Abstract: We answer a question of D. A. Hill in the negative by providing two local artinian rings $ R$ and $ S$ such that $ R$ is right serial but the left indecomposable injective $ R$-module is not uniserial, and that $ S$ is not right serial but the left indecomposable injective $ S$-module is uniserial. Moreover $ R$ possesses a Morita duality but fails to have self-duality.


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  • [1] F. W. Anderson and K. R. Fuller, Rings and categories of modules, Springer-Verlag, Berlin/ New York, 1974. MR 0417223 (54:5281)
  • [2] G. Azumaya, A duality theory for injective modules, Amer. J. Math. 81 (1959), 249-278. MR 0106932 (21:5662)
  • [3] V. Dlab and C. M. Ringel. Balanced rings, Lecture Notes in Math. 246, Springer-Verlag, Berlin/New York, 1972. MR 0340344 (49:5099)
  • [4] P. Dowbor, C. M. Ringel and D. Simson, Hereditary artinian rings of finite representation type, Lecture Notes in Math. 832, Springer-Verlag, Berlin/New York, 1980. MR 607156 (82i:16029)
  • [5] D. A. Hill, Rings whose indecomposable injective modules are uniserial, Canad. J. Math. 34 (1982), 797-805. MR 672676 (84h:16016)
  • [6] J. Kraemer, Characterizations of the existence of (Quasi-) self-duality for complete tensor rings, Algebra Berichte 56, Verlag Reinhard Fischer, München 1987. MR 908292 (89e:16039)
  • [7] K. Morita, Duality for modules and its applications to the theory of rings with minimum condition, Tokyo Kyoiku Daigaku, Ser. A6 (1958), 83-142. MR 0096700 (20:3183)
  • [8] B. J. Müller, On Morita duality, Canad. J. Math. 21 (1969), 1338-1347. MR 0255597 (41:258)
  • [9] -, The structure of quasi-Frobenius rings, Canad. J. Math. 26 (1974), 1141-1151. MR 0379577 (52:482)
  • [10] A. H. Schofield, Representations of rings over skew fields, London Math. Soc. Lecture Note Series 92, Cambridge Univ. Press 1985. MR 800853 (87c:16001)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0979222-2
Keywords: Uniserial modules, serial rings, Morita duality
Article copyright: © Copyright 1989 American Mathematical Society

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