The theory of 𝑄-rings

Matlis, Eben

doi:10.1090/S0002-9947-1974-0340241-4

The theory of $Q$-rings
HTML articles powered by AMS MathViewer

by Eben Matlis PDF

Trans. Amer. Math. Soc. 187 (1974), 147-181 Request permission

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

Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
C. U. Jensen, On the structure of $\textrm {Ext}_{R}^{1}(A,\,R)$, Rings, modules and radicals (Proc. Colloq., Keszthely, 1971) Colloq. Math. Soc. János Bolyai, Vol. 6, North-Holland, Amsterdam, 1973, pp. 215–226. MR 0360554
C. U. Jensen, On $\textrm {Ext}^{1}_{R}(A,\,R)$ for torsion-free $A$, Bull. Amer. Math. Soc. 78 (1972), 831–834. MR 311657, DOI 10.1090/S0002-9904-1972-13052-0
Daniel M. Kan and George W. Whitehead, On the realizability of singular cohomology groups, Proc. Amer. Math. Soc. 12 (1961), 24–25. MR 123319, DOI 10.1090/S0002-9939-1961-0123319-6
Eben Matlis, Divisible modules, Proc. Amer. Math. Soc. 11 (1960), 385–391. MR 116044, DOI 10.1090/S0002-9939-1960-0116044-8
Eben Matlis, Some properties of Noetherian domains of dimension one, Canadian J. Math. 13 (1961), 569–586. MR 131437, DOI 10.4153/CJM-1961-046-x
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, Rings of type I, J. Algebra 23 (1972), 76–87. MR 306185, DOI 10.1016/0021-8693(72)90046-4
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
D. G. Northcott, A general theory of one-dimensional local rings, Proc. Glasgow Math. Assoc. 2 (1956), 159–169. MR 76748, DOI 10.1017/S204061850003327X
R. J. Nunke and J. J. Rotman, Singular cohomology groups, J. London Math. Soc. 37 (1962), 301–306. MR 140557, DOI 10.1112/jlms/s1-37.1.301
Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. II, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0120249, DOI 10.1007/978-3-662-29244-0
Friedrich Karl Schmidt, Mehrfach perfekte Körper, Math. Ann. 108 (1933), no. 1, 1–25 (German). MR 1512831, DOI 10.1007/BF01452819