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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 + 0 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 K,-free Abelian groups of cardinality K1 in ZFC and assuming the weak Continuum Hypothesis.
Gobel, R. & Goldsmith, B. (1993). The Kaplansky Test Problems - An approach via radicals. Journal of Pure and Applied Algebra, vol. 99, pg. 331-344. doi.org/10.1016/0022-4049(93)E0182-4