### Reflexive subgroups of the Baer-Specker group and Martin's axiom

by Goebel and Shelah. [GbSh:727]

Proc Perth Conference 2000 ``AGRAM'', 2001

In two recent papers we answered a question raised in the book
by Eklof and Mekler (p. 455, Problem 12) under the set theoretical
hypothesis of diamondsuit_{aleph_1} which holds in many models
of set theory, respectively of the special continuum hypothesis
(CH). The objects are reflexive modules over countable principal
ideal domains R, which are not fields. Following H. Bass, an
R-module G is reflexive if the evaluation map sigma :
G ---> G^{**} is an isomorphism. Here G^*= Hom (G,
R) denotes the dual module of G . We 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 mentioned above. Is CH a necessary
condition to find `nasty' reflexive modules? In the last part of
this paper we will show (assuming the existence of supercompact
cardinals) that large reflexive modules always have large
summands. So at least being essentially indecomposable needs an
additional set theoretic assumption. However the assumption need not
be CH as shown in the first part of this paper. We will use Martin's
axiom to find reflexive modules with the above decomposition which
are submodules of the Baer-Specker module R^omega .

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