# Sh:716

- Göbel, R., & Shelah, S. (2001).
*Decompositions of reflexive modules*. Arch. Math. (Basel),**76**(3), 166–181. arXiv: math/0003165 DOI: 10.1007/s000130050557 MR: 1816987 -
Abstract:

We continue [GbSh:568], proving a stronger result under the special continuum hypothesis (CH). The original question of Eklof and Mekler related to dual abelian groups. We want to find a particular example of a dual group, which will provide a negative answer to the question. In order to derive a stronger and also more general result we will concentrate on reflexive modules over countable principal ideal domains R. Following H. Bass, an R-module G is reflexive if the evaluation map \sigma:G\longrightarrow G^{**} is an isomorphism. Here G^*={\rm Hom} (G,R) denotes the dual group of G. Guided by classical results the question about the existence of a reflexive R-module G of infinite rank with G\not\cong G\oplus R is natural. We will use a theory of bilinear forms on free R-modules which strengthens our algebraic results in [GbSh:568]. Moreover we want to apply a model theoretic combinatorial theorem from [Sh:e] which allows us to avoid the weak diamond principle. This has the great advantage that the used prediction principle is still similar to the diamond, but holds under CH. - Version 2000-02-19_11 (21p) published version (16p)

Bib entry

@article{Sh:716, author = {G{\"o}bel, R{\"u}diger and Shelah, Saharon}, title = {{Decompositions of reflexive modules}}, journal = {Arch. Math. (Basel)}, fjournal = {Archiv der Mathematik}, volume = {76}, number = {3}, year = {2001}, pages = {166--181}, issn = {0003-889X}, mrnumber = {1816987}, mrclass = {20K20 (03E35 13C13 20K30)}, doi = {10.1007/s000130050557}, note = {\href{https://arxiv.org/abs/math/0003165}{arXiv: math/0003165}}, arxiv_number = {math/0003165} }