transactions of the american mathematical society Volume 322. Number 2. December 1990 BAER MODULES OVER DOMAINS PAUL C. EKLOF, LASZLO FUCHS, AND SAHARON SHELAH Abstract. For a commutative domain R with 1, an Ä-module B is called a Baer module if E\ilR(B , T) = 0 for all torsion K-modules T. The structure of Baer modules over arbitrary domains is investigated, and the problem is reduced to the case of countably generated Baer modules. This requires a general version of the singular compactness theorem. As an application we show that over h- local Prüfer domains, Baer modules are necessarily projective. In addition, we establish an independence result for a weaker version of Baer modules. 0. Introduction In his seminal paper [B] on mixed abelian groups, R. Baer proved that a countable abelian group B had to be free if Extz(ß, T) = 0 for all torsion abelian groups T. The problem of determining the uncountable groups B with this property turned out to be extremely difficult. Only 30 years later was it settled by Griffith [Gf] who showed that B had to be free, no matter what its cardinality was. Nunke [N] and Grimaldi [Gm] generalized the results to modules over Dedekind domains. Recently, Göbel [Gö] showed that the results extend to torsion theories over Dedekind domains. The problem of characterizing Baer modules B over arbitrary domains R (i.e., A-modules B with ExtR(B, T) = 0 for all torsion /î-modules T) was raised by Kaplansky [K]. He established two useful lemmas that served as a starting point for the paper of two of the present authors [EF] in which a new approach was used to show that Baer modules over arbitrary valuation domains had to be free. While particular properties of modules over valuation domains were utilized to settle the case of countable rank, transfinite induc- tion was needed for higher ranks. For regular cardinals, a lemma was proved which—in a more general form—will serve as a crucial lemma in the present paper. For singular cardinals, a version of Shelah's compactness theorem was used. The point of departure for the present paper is the observation that the tools developed in [EF] can be refined to deal with Baer modules over arbitrary do- Received by the editors December 13, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 13C05, 13G05, 13L05. Key words and phrases. Baer module, flat and projective modules, continuous ascending chains, Prüfer domains, constructibility, Proper Forcing Axiom, singular compactness. This research was partially supported by NSF Grants DMS-8600376 and DMS-8620379. ©1990 American Mathematical Society 0002-9947/90 $1.00+ $.25 per page 547 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use