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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
MathSciNet review: 0340241
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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.


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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