THE COMPLEXITY OF THE EMBEDDABILITY RELATION BETWEEN TORSION-FREE ABELIAN GROUPS OF UNCOUNTABLE SIZE

We prove that for every uncountable cardinal kappa such that kappa(<kappa) = kappa, the quasi-order of embeddability on the kappa-space of kappa-sized graphs Borel reduces to the embeddability on the kappa-space of kappa-sized torsion-free abelian groups. Then we use the same techniques to prove that the former Borel reduces to the embeddability relation on the kappa-space of kappa-sized R-modules, for every S-cotorsion-free ring R of cardinality less than the continuum. As a consequence we get that all the previous are complete Sigma(1)(1) quasi-orders.