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Module-theoretic characterizations of the ring of finite fractions of a commutative ring. (English) Zbl 1496.13004

Let \(D\) be an integral domain and \(K\) its field of quotients. It is well-known, that \(D\) is integrally closed in \( K\) if and only if the polynomial ring \(D[X]\) is integrally closed in its field of quotients \(K(X)\). When considering a commutative ring \(R\) with zero-divisors, traditionally, researchers use the total ring of quotients \(T (R)\) (i.e., the localization of \(R\) at the set of all regular elements of \(R\)) to replace the role of the field of quotients of a domain. However, it is worth noting that some classical theorems on domains cannot be generalized to commutative rings perfectly in terms of \(T(R)\). For instance J. W. Brewer et al. [J. Algebra 58, 217–226 (1979; Zbl 0446.13004)], showed that it is not always the case that \(R[X]\) is integrally closed in \(T(R[X]) \) when \(R\) is integrally closed in \(T(R)\). To remedy such a problem, Lucas introduced another extension of \(R\), larger that \(T(R)\), \(Q_0(R)\) called the ring of finite fractions of \(R\) and he proved that a reduced ring \(R\) is integrally closed in \(Q_0(R)\) if and only if \(R[X]\) is integrally closed in \(T(R[X])\). Thus, when dealing with some relative properties of commutative rings corresponding to domains, sometimes the behavior of \(Q_0(R)\) is much better than that of \(T(R)\).
Recall that an ideal \(I\) of \(R\) is said to be semiregular if there exists some finitely generated subideal \(I_0\) of \(I\) such that ann\((I_0)=0\). Note that the set of finitely generated semiregular ideals of \(R\), denoted by \(\mathcal{Q}\), is a multiplicative system of ideals of \(R\), and \[ Q_0(R) := \{ \alpha \in T (R[X]) \mid \mbox{ there exists some } I \in \mathcal{Q} \mbox{ such that } \alpha I \subseteq R \}. \] In the present paper, the authors discuss module-theoretic properties of \(Q_0(R)\) and introduce Lucas modules and DQ rings: a \(\mathcal{Q}\)-torsion-free \(R\)-module \(M\) is called a Lucas module if Ext\(^1_R(R/J, M) = 0 \) for any \(J \in \mathcal{Q}\); \(R\) is called a DQ ring if every ideal of \(R\) is a Lucas module.
They prove that, if the small finitistic dimension of \(R\) is zero, then \(R\) is a DQ ring. Moreover, in terms of a trivial extension, they construct a total ring of quotients of the type \(R = D \varpropto H\), for particular domain \(D\) and \(D\)-module \(H\), which is not a DQ ring. Thus, in this case, the small finitistic dimension of \(R\) is not zero. This provides a negative answer to an open problem posed by P.-J. Cahen et al. [in: Commutative algebra. Recent advances in commutative rings, integer-valued polynomials, and polynomial functions. Based on mini-courses and a conference on commutative rings, integer-valued polynomials and polynomial functions, Graz, Austria, December 16–18 and December 19–22, 2012. New York, NY: Springer. 353–375 (2014; Zbl 1327.13002)].

MSC:

13A15 Ideals and multiplicative ideal theory in commutative rings
13C99 Theory of modules and ideals in commutative rings
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References:

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