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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Krull-Schmidt fails for serial modules


Author: Alberto Facchini
Journal: Trans. Amer. Math. Soc. 348 (1996), 4561-4575
MSC (1991): Primary 16D70, 16S50, 16P60
DOI: https://doi.org/10.1090/S0002-9947-96-01740-0
MathSciNet review: 1376546
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Abstract: We answer a question posed by Warfield in 1975: the Krull-Schmidt Theorem does not hold for serial modules, as we show via an example. Nevertheless we prove a weak form of the Krull-Schmidt Theorem for serial modules (Theorem 1.9). And we show that the Grothendieck group of the class of serial modules of finite Goldie dimension over a fixed ring $R$ is a free abelian group.


References [Enhancements On Off] (What's this?)

  • 1. R. Camps and W. Dicks, On semilocal rings, Israel J. Math. 81 (1993), 203--211. MR 94m:16027
  • 2. A. Facchini, D. Herbera, L. S. Levy and P. Vámos, Krull-Schmidt fails for artinian modules, Proc. Amer. Math. Soc. 123 (1995), 3587--3592. MR 96b:16020
  • 3. A. Facchini and L. Salce, Uniserial modules: sums and isomorphisms of subquotients, Comm. Algebra 18(2) (1990), 499--517. MR 91h:16005
  • 4. D. Herbera and A. Shamsuddin, Modules with semi-local endomorphism ring, Proc. Amer. Math. Soc. 123 (1995), 3593--3600. MR 96b:16014
  • 5. I. Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc. 66 (1949), 464--491. MR 11:155b
  • 6. B. Stenström, Rings of Quotients, Springer-Verlag, Berlin, 1975.
  • 7. K. Varadarajan, Dual Goldie dimension, Comm. Algebra 7 (1979), 565--610. MR 80d:16014
  • 8. R. B. Warfield, Serial rings and finitely presented modules, J. Algebra 37 (1975), 187--222. MR 53:5663

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Additional Information

Alberto Facchini
Affiliation: Dipartimento di Matematica e Informatica, Università di Udine, 33100 Udine, Italy
Email: facchini@dimi.uniud.it

DOI: https://doi.org/10.1090/S0002-9947-96-01740-0
Received by editor(s): August 4, 1995
Additional Notes: Partially supported by Ministero dell’Università e della Ricerca Scientifica e Tecnologica (Fondi 40% e 60%), Italy. This author is a member of GNSAGA of CNR
Article copyright: © Copyright 1996 American Mathematical Society

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