Reflexive subgroups of the Baer-Specker group and Martin's axiom (Q2738753)

In two papers [Arch. Math. 76, No. 3, 166-181 (2001; Zbl 1014.03054) and Math. Z. 237, No. 3, 547-559 (2001; Zbl 0988.20044)] the authors of the paper under review answered a question raised in the book by \textit{P. C. Eklof} and \textit{A. H. Mekler} [``Almost free modules. Set-theoretic methods'', North-Holland, Amsterdam (1990; Zbl 0718.20027), see page 455, Problem 12] under the set theoretical hypothesis of \(\lozenge_{\aleph_1}\) which holds in many models of set theory, respectively of special continuum hypothesis (CH). The objects are reflexive modules over countable principal ideal domains \(R\), which are not fields. Following \textit{H. Bass} [Trans. Am. Math. Soc. 95, 466-488 (1960; Zbl 0094.02201)] an \(R\)-module \(G\) is said to be reflexive if the evaluation map \(\sigma\colon G\to G^{**}\) is an isomorphism, where \(G^*=\Hom(G,R)\) denotes the dual module of \(G\). They proved the existence of reflexive \(R\)-modules \(G\) of infinite rank with (*) \(G\not\cong G\oplus R\), which provide (even essentially indecomposable) counter examples to the question above of Eklof and Mekler: Is CH a necessary condition to find `nasty' reflexive modules?NEWLINENEWLINENEWLINEThe results of the paper under review are obtained for Abelian groups and the authors mention that it is an easy exercise to replace the ground ring \(\mathbb{Z}\) by any countable principal ideal domain which is not a field. It is also mentioned that the authors could work with one prime only. An Abelian group \(G\) is said to be a dual if \(G\cong D^*\) for some Abelian group \(D\). In the third section of this paper the authors show: (Theorem 1.1 or Theorem 3.1) If \(\kappa\) is a supercompact cardinal and \(H\) is a dual group of cardinality \(\geq\kappa\), then there is a direct summand \(H'\) of \(H\) with \(\chi\leq|H'|<\kappa\) for any cardinal \(\chi<\kappa\). As reflexive groups are dual this theorem (assuming the existence of supercompact cardinals) shows that large reflexive modules always have large summands. So at least being essentially indecomposable needs additional set theoretic assumptions. However the assumption need not be CH as shown in the first part of this paper. As CH implies Martin's axiom, the main result of this paper (Theorem 1.2) gives a new proof of the existence of reflexive groups as in [Arch. Math. (loc. cit.)]. The authors in Theorem 1.2 use Martin's axiom to find reflexive modules with the above decomposition (*) which are submodules of the Baer-Specker module \(R^\omega\). New algebraic and combinatorial methods and some old techniques from earlier papers like [\textit{R. Göbel, B. Wald}, Math. Z. 172, 107-121 (1980; Zbl 0455.20038)] or [\textit{M. Dugas, J. Irwin, S. Khabbaz}, Q. J. Math., Oxf. II. Ser. 39, No. 154, 201-211 (1988; Zbl 0663.20058)] have been used to prove Theorem 1.2.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00043].