Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

Rings with property $ D$


Author: Eben Matlis
Journal: Trans. Amer. Math. Soc. 170 (1972), 437-446
MSC: Primary 13G05
DOI: https://doi.org/10.1090/S0002-9947-1972-0306186-9
MathSciNet review: 0306186
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: An integral domain is said to have property $ {\text{D}}$ if every torsion-free module of finite rank is a direct sum of modules of rank one. In recent papers the author has given partial solutions to the problem of finding all rings with this property. In this paper the author is finally able to show that an integrally closed integral domain has property $ {\text{D}}$ if and only if it is the intersection of at most two maximal valuation rings.


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

  • [1] H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton, N. J., 1956. MR 17, 1040. MR 0077480 (17:1040e)
  • [2] I. Kaplansky, Modules over Dedekind rings and valuation rings, Trans. Amer. Math. Soc. 72 (1952), 327-340. MR 13, 719. MR 0046349 (13:719e)
  • [3] E. Matlis, Injective modules over Noetherian rings, Pacific J. Math. 8 (1958), 511-528. MR 20 #5800. MR 0099360 (20:5800)
  • [4] -, Injective modules over Prüfer rings, Nagoya Math. J. 15 (1959), 57-69. MR 22 #725. MR 0109840 (22:725)
  • [5] -, Cotorsion modules, Mem. Amer. Math. Soc. No. 49 (1964), 66 pp. MR 31 #2283. MR 0178025 (31:2283)
  • [6] -, Decomposable modules, Trans. Amer. Math. Soc. 125 (1966), 147-179. MR 34 #1349. MR 0201465 (34:1349)
  • [7] -, The decomposability of torsion free modules of finite rank, Trans. Amer. Math. Soc. 134 (1968), 315-324. MR 37 #6317. MR 0230757 (37:6317)
  • [8] -, Rings of type I, J. Algebra 23 (1972). MR 0306185 (46:5312)
  • [9] -, Local $ D$-rings, Math. Z. 124 (1972), 266-272. MR 0291161 (45:255)
  • [10] M. Nagata, Local rings, Interscience Tracts in Pure and Appl. Math., no. 13, Interscience, New York, 1962. MR 27 #5790. MR 0155856 (27:5790)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 13G05

Retrieve articles in all journals with MSC: 13G05


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0306186-9
Keywords: Maximal valuation ring, torsion-free module, direct sum decomposition
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society