Rings with property 𝐷

Matlis, Eben

doi:10.1090/S0002-9947-1972-0306186-9

Rings with property $D$
HTML articles powered by AMS MathViewer

by Eben Matlis PDF

Trans. Amer. Math. Soc. 170 (1972), 437-446 Request permission

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

Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
Irving Kaplansky, Modules over Dedekind rings and valuation rings, Trans. Amer. Math. Soc. 72 (1952), 327–340. MR 46349, DOI 10.1090/S0002-9947-1952-0046349-0
Eben Matlis, Injective modules over Noetherian rings, Pacific J. Math. 8 (1958), 511–528. MR 99360
Eben Matlis, Injective modules over Prüfer rings, Nagoya Math. J. 15 (1959), 57–69. MR 109840
Eben Matlis, Cotorsion modules, Mem. Amer. Math. Soc. 49 (1964), 66. MR 178025
Eben Matlis, Decomposable modules, Trans. Amer. Math. Soc. 125 (1966), 147–179. MR 201465, DOI 10.1090/S0002-9947-1966-0201465-5
Eben Matlis, The decomposability of torsion free modules of finite rank, Trans. Amer. Math. Soc. 134 (1968), 315–324. MR 230757, DOI 10.1090/S0002-9947-1968-0230757-0
Eben Matlis, Rings of type I, J. Algebra 23 (1972), 76–87. MR 306185, DOI 10.1016/0021-8693(72)90046-4
Eben Matlis, Local $D$-rings, Math. Z. 124 (1972), 266–272. MR 291161, DOI 10.1007/BF01113920
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