The theory of 𝑄-rings

Abstract

An integral domain R with quotient field Q is defined to be a Q -ring if Ext R 1 ⁡ ( Q , R ) ≅ Q \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 Hom R ( A , R ) = 0 {\operatorname {Hom}_R}(A,R) = 0 and Ext R 1 ⁡ ( A , R ) ≅ Q \operatorname {Ext}_R^1(A,R) \cong Q . If A is such an R -module and t ( A ) t(A) is its torsion submodule, then it is proved that A / t ( A ) A/t(A) necessarily has rank one. There are only three kinds of Q -rings, namely, Q 0 - , Q 1 - {Q_0}{\text {-}},{Q_1}{\text {-}} , or Q 2 {Q_2} -rings. These are described by the fact that if R is a Q -ring, then K = Q / R 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 ⊗ R Q 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.