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Direct sums of local torsion-free abelian groups


Author: David M. Arnold
Journal: Proc. Amer. Math. Soc. 130 (2002), 1611-1617
MSC (2000): Primary 20K15, 20K25
DOI: https://doi.org/10.1090/S0002-9939-01-06246-3
Published electronically: November 15, 2001
MathSciNet review: 1887006
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Abstract: The category of local torsion-free abelian groups of finite rank is known to have the cancellation and $n$-th root properties but not the Krull-Schmidt property. It is shown that 10 is the least rank of a local torsion-free abelian group with two non-equivalent direct sum decompositions into indecomposable summands. This answers a question posed by M.C.R. Butler in the 1960's.


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

  • [Arnold 72] Arnold, D. A duality for torsion-free modules of finite rank over a discrete valuation ring, Proc. Lond. Math. Soc. (3) 24 (1972), 204-216.
  • [Arnold 82] Arnold, D. Finite Rank Torsion-Free Abelian Groups and Rings, Lect. Notes in Math. 931, Springer-Verlag, New York, 1982. MR 84d:20002
  • [Arnold 00] Arnold, D. Abelian Groups and Representations of Finite Partially Ordered Sets, CMS Books in Mathematics, Springer-Verlag, New York, 2000. MR 2001g:16030
  • [Arnold 01] Arnold, D. Direct sum decompositions of torsion-free abelian groups of finite rank, Abelian Groups, Rings, and Modules (Proc. of 2000 Perth Conf.), Cont. Math., AMS, Providence, Rhode Island, 2001, 65-74. CMP 2001:09
  • [Arnold Dugas 00] Arnold, D. and Dugas, M. Co-purely indecomposable modules over a discrete valuation ring, J. Pure and Appl. Alg. 161 (2001), 1-12.
  • [Arnold Lady 75] Arnold, D. and Lady, E.L. Endomorphism rings and direct sums of torsion-free abelian groups, Trans. Amer. Math. Soc. 211 (1975), 225-237. MR 54:5370
  • [Fuchs 73] Fuchs, L. Infinite Abelian Groups, Vol. II, Academic Press, New York, 1973. MR 50:2362
  • [Goldsmith May 99] Goldsmith, B. and May, W. The Krull-Schmidt problem for modules over valuation domains, J. Pure and Appl. Alg. 140 (1999), 57-63. MR 2000d:13016
  • [Lady 75] Lady, E.L. Nearly isomorphic torsion-free abelian groups, J. Alg. 35 (1975), 235-238. MR 51:5801
  • [Lady 77] Lady, E.L. Splitting fields for torsion-free modules over discrete valuation rings, J. Alg. 49 (1977), 261-275. MR 58:22039
  • [Walker 64] Walker, E.A. Quotient categories and quasi-isomorphisms of abelian groups, Proc. Colloq. Abelian Groups, Budapest, 1964, 147-162. MR 31:2327
  • [Warfield 72] Warfield, R.B. Jr. Exchange rings and decompositions of modules, Math. Ann. 199 (1972), 31-36. MR 48:11218

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Additional Information

David M. Arnold
Affiliation: Department of Mathematics, Baylor University, Waco, Texas 76798-7328
Email: David_Arnold@baylor.edu

DOI: https://doi.org/10.1090/S0002-9939-01-06246-3
Keywords: Krull-Schmidt groups, direct sum decompositions, local torsion-free abelian groups
Received by editor(s): October 4, 2000
Received by editor(s) in revised form: January 8, 2001
Published electronically: November 15, 2001
Additional Notes: This research was supported, in part, by the Baylor University Summer Sabbatical Program
Communicated by: Stephen D. Smith
Article copyright: © Copyright 2001 American Mathematical Society

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