The kaplansky test problems — an approach via radicals Original Research Article

The existence of non-free, k-free Abelian groups and modules (over some non-left perfect rings R) having prescribed endomorphism algebra is established within ZFC + ◊ set theory. The principal technique used exploits free resolutions of non-free R-modules X and is similar to that used previously by Griffith and Eklof; much stronger results than have been obtained heretofore are obtained by coding additional information into the module X. As a consequence we can show, inter alia, that the Kaplansky Test Problems have negative answers for strongly aleph, Hebrew1-free Abelian groups of cardinality aleph, Hebrew1 in ZFC and assuming the weak Continuum Hypothesis.