Ideals and overrings of divided domains (Q2920181)

The paper under review establishes new properties of \textit{divided integral domains}. Divided integral domains were introduced by \textit{D. E. Dobbs} in [Pac. J. Math. 67, 353--363 (1976; Zbl 0326.13002)]. An integral domain \(A\) is said to be divided if each \(P\in\text{Spec}(A)\) satisfies \(P=PA_P\); that is, if each element of \(A\setminus P\) divides each element of \(P\), with quotient in \(A\). Dobbs showed that a divided integral domain is a quasilocal going-down domain; that is, each of its overring extensions has the going-down property (Proposition 2.1 in [Zbl 0326.13002]). The author also considers \textit{divided rings}, as introduced by \textit{A. Badawi} in [Commun. Algebra 27, No. 3, 1465--1474 (1999; Zbl 0923.13001)]. These are rings in which each (principal) ideal is comparable to any prime ideal. A commutative ring is called \textit{treed} if any two incomparable prime ideals are coprime. When the ring is quasilocal, being treed means that the prime spectrum of the ring is linearly ordered under inclusion.NEWLINENEWLINEFrom now on all rings will be commutative and unital. The author uses the following notation: Given a ring morphism \(f: A\rightarrow B\), let \(S_f\) be the multiplicatively closed subset \(\{s\in A\mid f(s)\;\text{is\;a\;unit\;in\;}B\}\) and let \(\text{Spec}(B\mid A)\) be the image of the corresponding spectral morphism \(\text{Spec}(B)\rightarrow\text{Spec}(A)\). If \(P\in\text{Spec}(A)\), let \(P^{\downarrow}:=\text{Spec}(A_P\mid A)\) be the generalization of \(P\), i.e., \(Q\in P^{\downarrow}\) if and only if \(Q\subseteq P\). Let \(\text{Max}(A)\) be the set of maximal ideals of \(A\) and let \(\text{Tot}(A)\) be its total quotient ring. If \(I\) is an ideal of \(A\), let \(\Lambda(I)\) be the multiplicatively closed subset of \(A\) consisting of elements that become a unit in \(\text{Tot}(A/I)\). Also let \(\text{Ass}_f(I)\) be the set of all \textit{Bourbaki associated prime ideals} of \(I\). A prime ideal \(P\) is in \(\text{Ass}_f(I)\) if \(P\in\text{Min}(I: x)\) for some \(x\in A\). Finally, let \(\text{Ass}(I)\) be the \textit{Krull associated prime ideals} of \(I\). A prime ideal of \(A\) is in \(\text{Ass}(I)\) if and only if it is a union of some elements of \(\text{Ass}_f(I)\) (cf. Lemma 2.1 in [\textit{L. Fuchs} et al., Trans. Am. Math. Soc. 357, No. 7, 2771--2798 (2005; Zbl 1066.13003)]).NEWLINENEWLINEWith this notation, the key result of this paper states that if \(f:A\rightarrow B\) is a going-down ring morphism, where \(A\) is a quasilocal treed ring, then: (a) \(S_f\) is a saturated multiplicatively closed subset of \(A\), \(P_f:=A\setminus S_f\in\text{Spec}(A)\) is such that \(\bigcap[As\mid s\in S_f]\subseteq P_f=\bigcup[A\cap N\mid N\in\text{Max}(B)]\) and \(\text{Spec}(B\mid A)=(P_f)^{\downarrow}\). If in addition \(A\) is a divided ring, then \(\bigcap[As\mid s\in S_f]=P_f\). (b) There is a factorization \(A\rightarrow A_{P_f}\rightarrow B\), where \(A_{P_f}\rightarrow B\) has the lying-over property and the going-down property. (c) In case \(A\) is an integral domain and \(f\) is a flat epimorphism, then \(B\simeq\bigcap[B_P\mid P\in\text{Spec}(B\mid A)]\), \(P_f=\bigcup[P\mid P\in\text{Spec}(B\mid A)]\) and \(B=A_{P_f}\). The author then uses this result to show that if \(f: A\rightarrow B\) is an injective going-down ring morphism, where \(A\) is a quasilocal treed domain, then \(A\rightarrow A_{P_f}\) is the Morita maximal flat epimorphic subextension of \(A\rightarrow B\) (cf. [\textit{K. Morita}, Math. Z. 120, 25--40 (1971; Zbl 0203.34203)]). The author also obtains results on ideals of a divided ring. One main result obtained is that if \(R\) is a divided ring and \(I\neq R\) is an ideal of \(R\), then \(\text{Ass}(I)=\text{Ass}_f(I)=V(I)\cap(R\setminus\Lambda(I))^{\downarrow}\) and this set is (Zariski) compact in \(\text{Spec}(R)\).NEWLINENEWLINEThe author then considers conductor overrings \((I: I)\) associated to a nonzero ideal \(I\) of an integral domain \(A\). Let \(\pi: A\rightarrow (I: I)\) be the natural map. The main result proved states that if \(A\) is a divided domain and \(I\neq A\) is a nonzero ideal of \(A\), then \(\text{Spec}((I: I)\mid A)=(R\setminus\Lambda(I))^{\downarrow}\) and if \(^t\pi:\text{Spec}(I: I)\rightarrow\text{Spec}(A)\) is the spectral morphism corresponding to \(\pi\), then \(^t\pi(\text{Max}((I: I)))\subseteq V(I)\cap(R\setminus\Lambda(I))^{\downarrow}=\text{Ass}_f(I)=\text{Ass}(I)\) and \((R\setminus\Lambda(I))\in{^t\pi}(\text{Max}((I: I)))\). When \(I\) is a primary ideal, the author concludes that \({^t\pi}(\text{Max}((I: I)))=\{(R\setminus\Lambda(I))\}\).NEWLINENEWLINEAmong several other results obtained, it is proved: (1) In a divided integral domain \(R\) every Goldman prime ideal \(P\) is a \(g\)-ideal, that is, \(P^{\downarrow}\) is an open subset of \(\text{Spec}(R)\), necessarily of the form \(D(a)\), where \(a\in R\) is nonzero. (2) \textit{R. Gilmer's} characterization of a nonzero unbranched prime ideal of a Prüfer domain (Theorem 17.3 in [Multiplicative ideal theory. Rev. ed. Kingston: Queen's University (1992; Zbl 0804.13001)]) is still valid in a divided integral domain. (3) A divided domain is an \(\Omega\)-domain (in the sense of \textit{M. Fontana} and \textit{E. Houston} [J. Pure Appl. Algebra 163, No. 2, 173--192 (2001; Zbl 1094.13500)]), if and only if it is a \(QQR\)-domain and each nonzero prime ideal is a \(G\)-ideal.NEWLINENEWLINEThe paper ends by giving conditions for the sequence \(\{\text{Ass}(I^n)\}_{n>0}\) to be stationary for an ideal \(I\) of a divided domain. This question has been the subject of many papers in the noetherian context.