Indecomposable modules over Nagata valuation domains

Arnold, D.; Dugas, M.

doi:10.1090/S0002-9939-1994-1239795-3

Indecomposable modules over Nagata valuation domains
HTML articles powered by AMS MathViewer

by D. Arnold and M. Dugas PDF

Proc. Amer. Math. Soc. 122 (1994), 689-696 Request permission

Abstract:

For a discrete valuation ring R, let ${\text {fr}}(R)$ be the supremum of the ranks of indecomposable finite rank torsion-free R-modules. Then ${\text {fr}}(R) = 1,2,3$, or $\infty$. A complete list of indecomposables is given if ${\text {fr}}(R) \leq 3$, in which case R is known to be a Nagata valuation domain.

References

David M. Arnold, A duality for torsion-free modules of finite rank over a discrete valuation ring, Proc. London Math. Soc. (3) 24 (1972), 204–216. MR 292813, DOI 10.1112/plms/s3-24.2.204
David M. Arnold, Finite rank torsion free abelian groups and rings, Lecture Notes in Mathematics, vol. 931, Springer-Verlag, Berlin-New York, 1982. MR 665251
László Fuchs, Infinite abelian groups. Vol. I, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. MR 0255673
E. L. Lady, On classifying torsion free modules over discrete valuation rings, Abelian group theory (Proc. Second New Mexico State Univ. Conf., Las Cruces, N.M., 1976) Lecture Notes in Math., Vol. 616, Springer, Berlin, 1977, pp. 168–172. MR 0498527
E. L. Lady, A seminar on splitting rings for torsion free modules over Dedekind domains, Abelian group theory (Honolulu, Hawaii, 1983) Lecture Notes in Math., vol. 1006, Springer, Berlin, 1983, pp. 1–48. MR 722612, DOI 10.1007/BFb0103696
E. L. Lady, Splitting fields for torsion-free modules over discrete valuation rings. I, J. Algebra 49 (1977), no. 1, 261–275. MR 506213, DOI 10.1016/0021-8693(77)90285-X
Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155856
P. Ribenboim, On the completion of a valuation ring, Math. Ann. 155 (1964), 392–396. MR 164960, DOI 10.1007/BF01350748
Peter Vámos, Decomposition problems for modules over valuation domains, J. London Math. Soc. (2) 41 (1990), no. 1, 10–26. MR 1063538, DOI 10.1112/jlms/s2-41.1.10
Paolo Zanardo, Kurosch invariants for torsion-free modules over Nagata valuation domains, J. Pure Appl. Algebra 82 (1992), no. 2, 195–209. MR 1182938, DOI 10.1016/0022-4049(92)90120-5