Decompositions of reflexive modules (Q5937384)

If \(G\) is an arbitrary \(R\)-module, then \(G^* = \text{Hom}(G,R)\) denotes its dual module. The module \(G\) is called reflexive if the natural evaluation map \(\sigma :G \to G^{**}\) given by the formula \(\sigma (g)(\varphi) = \varphi (g)\), \(g\in G\), \(\varphi\in\text{Hom} (G,R)\), is an isomorphism. The authors prove under ZFC + CH that if \(R\) is a countable domain which is not a field, then there is a family of \(2^{\aleph_1}\) pair-wise non-isomorphic reflexive \(R\)-modules \(G\) of cardinality \(\aleph_1\) such that \(G\ncong R\oplus G\) (Theorem 1.3).