ON LOCALLY DIVIDED INTEGRAL DOMAINS AND CPI-OVERRINGS

It is proved that an integral domain R is locally divided if and only if each CPI-extension of B (Ira the sense of Boisen and Sheldon) is R-flat (equivalently, if and only if each CPI-extension of R is a localization of R) Thus, each CPI-extension of a locally divided domain is also locally divided. Treed domains are characterized by the going-down behavior of their CPI-extensions. A new class of (not necessarily treed) domains, called CPIclosed domains, is introduced. Examples include locally divided domains, quasilocal domains of Krull dimension 2, and qusilocal domains with the QQRproperty. The property of being CPI-closed behaves nicely with respect to the D + M construction, but is not a local property.


i. INTRODUCTION.
In [3], Boisen and Sheldon recently introduced the notion of a CPI-extension of a (commutative integral) domain. For the reader's convenience, we recall the definition of this type of overring and summarize salient results from [3] at the beginning of section 2. Boisen and Sheldon [3, p. 729] have noted that a proper CPI-extension may be integral. Indeed, by combining [6,Corollary 2.6] and [7, Proposition 2.1], it follows that each CPl-extension of a domain R is integral if and only if R is a quasilocal going-down ring, in the sense of [5].
Since a proper integral overring cannot be flat [14,Proposition 2], one might expect a rather different class of domains R to be characterized by the property of having each CPl-extension of R being R-flat. The main result of section 2, Theorem 2.4, establishes that the domains %hus characterized are the locally divided domains introduced in [6]. Prefer domains are perhaps the most natural examples of locally divided domains. It was shown in [6] that any locally divided domain is a (not necessarily quasilocal) going-down ring, and that the converse holds in the root-closed case. As byproducts, Proposition 2.3 establishes that a CPI-extension T of a domain R is R-flat (if and,) only if T is a localization (in the sense of "ring of fractions" in [h, p. 5?]) of R and Corollary 2.6 establishes that each CPI-extenslon of a locally divided domain is also locally divided. The final result in section 2 shows how going-down behavior of CPI-extensions serves to characterize treed domains.
Divided domains are precisely the domains which coincide with each of their CPI-extensions. More generally, section 3 is devoted to studying domains R which are "CPI-closed," in the sense that each CPI-extension of a CPI-extension of R is itself a CPl-extensiQn of R (A more useful characterization of CPl-closed domains is given in Proposition 3.2.) Examples of CPl-closed domains include the locally divided domains; the quasilocal domains of Krull dimension 2; and quasilocal domains with the QQR-property (see Corollary 3.3, Remark 3.5 and Proposition 3.7, respectively). Despite expectations raised by the first-cited example, the second family of examples illustrates that a CPl-closed domain need not be treed. The quasilocal, treed CPl-closed domains are characterized in Proposition 3.9 and Remark 3.10(a). As shown by Example 3.6, being CPl-closed is not a local property, since the ring of polynomials in two variables over a field is not CPl-closed. Examples of non-CPl-closed domains with arbitrary Krull dimension exceeding 1 then result from the D + M-construction (see Proposition 3.11). In Remarks 3.8 and 3.10(5), (c), we raise some open questions relating divided domains, domains with the QQR-property, certain types of CPlclosed domains, and the A-domains of Gilmer and Huckaba [i0].

LOCALLY DIVIDED DOMAINS.
As defined by Boisen and Sheldon [3], a CPl-extenslon of the domain R is an overring of R of the form R + P for some prime ideal P of R The terminology "CPI" stands for "complete pre-image," and is well chosen inasmuch as R + P is easily shown to be (canonically isomorphic to) the pullback, in the category of commutative rings with unit, of the diagram where the vertical map is the canonical surjection and the horizontal map is the inclusion. Reasoning as in [ii, Remark 3.9], we find that applying the contravariant functor Spec to the above diagram produces a pushout diagram in the (dual) category of afflne schemes. In particular, Spec(R + P) may be viewed set-theoretlcally as the quotient space of the disjoint union of Spec() and Sp&c(R/P) in which the prime P of is identified with the zero prime P/P of R/P We next summarize some more precise order-theoretlc information gleaned from [3, Section 2]. The method used in the proof of [ (2) For each prime P of R the CPl-extension R + P is R-flat; (3) For each prime P of R the CPl-extenslon R + P is a localization of R (4) For any comparable primes P c Q of R the corresponding CPl-extensions satisfy R + P c R + QRQ (5) For any comparable primes P c Q of R the containment P c PRQ holds; (6) For any comparable primes P c Q of R the containment P c RQ holds.
PROOF. (5) = (i): Establishing (i), i.e., that RQ is a divided domain for each maximal (equivalently, for each prime) ideal Q of R amounts to showing, for each prime P of R which is contained in Q .that PRQ coincides with (PRQ) (RQ)pR Q By the result cited in the proof of Lemma 2.2 (a), the latter prime is lust (PRQ) which, since RQ = , simplifies to e As P c Q also forces PRQ c P it is now clear that (5) = (i  Finally, observe that the implications (4) = (6) and (6) = (5) are straightforward, and the proof is complete. (2) R c T has the going-down property for each CPl-extension T of R PROOF. (i) = (2). Suppose that T R + P for some prime P of R and, in particular, R + QRQ c A To establish the reverse inclusion, reverse the roles of P and Q in the preceding argument. This completes the proof. REMARK 3.8. We have seen that the class of valuation domains may be properly extended in (at least) two ways: to the class of divided domains, and to the class of quasilocal dmains having the QQR-property. It is easily seen that the former of these "extended classes" is not contained in the latter. Indeed, Gilmer   (b) The second of the three mples in Remark 3.5 shows that the "treed" conclusion in Proposition 3.9 cannot be strengthened to "going-down ring," and hence certainly cannot be strengthened to "divided." Note that this example satisfies the hypothesis of Proposition 3.9 since it satisfies the ostensibly more stringent assumption of Corollary 3.4. We say "ostensibly," for we do not know (but doubt) whether, conversely, "the sum of any two CPl-extensions of the domain R is itself a CPl-extension of R" implies "the set of CPl-extensions of R is totally ordered by inclusion." Of course, the implication hold if dim(R) < 3 (c) Despite (b), one ca__n strengthen the hypothesis in Corollary 3.4 in order to get a "going-down" conclusion. Indeed, Paplck [12, Lemma 2.41] has shown, with the aid of [i0, Theorem 4], that if the set of all the overrlngs of a domain R is totally ordered by inclusion, then R is a quasilocal i-domaln (in the terminology of [13]) and, hence, R is a going-down ring. The crucial point is that such an R is a A-domain. (Following Gilmer