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)

 
 

 

The theory of $ Q$-rings


Author: Eben Matlis
Journal: Trans. Amer. Math. Soc. 187 (1974), 147-181
MSC: Primary 13D99
DOI: https://doi.org/10.1090/S0002-9947-1974-0340241-4
MathSciNet review: 0340241
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: An integral domain R with quotient field Q is defined to be a Q-ring if $ \operatorname{Ext}_R^1(Q,R) \cong Q$. It is shown that R is a Q-ring if and only if there exists an R-module A such that $ {\operatorname{Hom}_R}(A,R) = 0$ and $ \operatorname{Ext}_R^1(A,R) \cong Q$. If A is such an R-module and $ t(A)$ is its torsion submodule, then it is proved that $ A/t(A)$ necessarily has rank one. There are only three kinds of Q-rings, namely, $ {Q_0}{\text{-}},{Q_1}{\text{-}}$, or $ {Q_2}$-rings. These are described by the fact that if R is a Q-ring, then $ K = Q/R$ can only have 0, 1, or 2 proper h-divisible submodules. If H is the completion of R in the R-topology, then R is one of the three kinds of Q-rings if and only if $ H{ \otimes _R}Q$ is one of the three possible kinds of 2-dimensional commutative Q-algebras. Examples of all three kinds of Q-rings are produced, and the behavior of Q-rings under ring extensions is examined. General conditions are given for a ring not to be a Q-ring. As an application of the theory, necessary and sufficient conditions are found for the integral closure of a non-complete Noetherian domain to be a complete discrete valuation ring.


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] C. U. Jensen, On the structure on $ \operatorname{Ext}_R^1(A,R)$, Colloquia Math. Soc. János Bolyai, 6 Rings, modules and radicals, Keszthely (Hungary), (1971), 215-226. MR 0360554 (50:13002)
  • [3] -, On $ \operatorname{Ext}_R(A,R)$ for torsion-free A, Bull. Amer. Math. Soc. 78 (1972), 831-834. MR 0311657 (47:219)
  • [4] D. M. Kan and G. W. Whitehead, On the realizability of singular cohomology groups, Proc. Amer. Math. Soc. 12 (1961), 24-25. MR 23 #A647. MR 0123319 (23:A647)
  • [5] E. Matlis, Divisible modules, Proc. Amer. Math. Soc. 11 (1960), 385-391. MR 22 #6839. MR 0116044 (22:6839)
  • [6] -, Some properties of Noetherian domains of dimension 1, Canad. J. Math. 13 (1961), 569-586. MR 24 #A1288. MR 0131437 (24:A1288)
  • [7] -, Cotorsion modules, Mem. Amer. Math. Soc. No. 49 (1964). MR 31 #2283. MR 0178025 (31:2283)
  • [8] -, Decomposable modules, Trans. Amer. Math. Soc. 125 (1966), 147-179. MR 34 #1349. MR 0201465 (34:1349)
  • [9] -, Rings of type I, J. Algebra 23 (1972), 76-87. MR 0306185 (46:5312)
  • [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)
  • [11] D. G. Northcott, General theory of one dimensional local rings, Proc. Glasgow Math. Assoc. 2 (1956), 159-169. MR 17, 938. MR 0076748 (17:938c)
  • [12] R. J. Nunke and J. J. Rotman, Singular cohomology groups, J. London Math. Soc. 37 (1962), 301-306. MR 25 #3975. MR 0140557 (25:3975)
  • [13] O. Zariski and P. Samuel, Commutative algebra. Vol. II, University Series in Higher Math., Van Nostrand, Princeton, N. J., 1960. MR 22 #11006. MR 0120249 (22:11006)
  • [14] F. K. Schmidt, Mehrfach perfekte Körper, Math. Ann. 108 (1933), 1-25. MR 1512831

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 13D99

Retrieve articles in all journals with MSC: 13D99


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0340241-4
Keywords: $ \operatorname{Ext}_R^1(Q,R) \cong Q$, R-topology, completion, cotorsion, h-divisible, torsion, full ring of quotients, strongly compatible ring extensions, integral extensions, minimal prime ideals, Noetherian domains of Krull dimension 1
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society