A module M over a ring R is ?o-projective, ?o a cardinal, if M is projective relative to all exact sequence of R-modules 0 a?’ A a?’ B a?’ C a?’ 0 such that C has a generating set of cardinality less than ?o. A structure theorem for ?o-projective modules over Dedekind domains is proven, and the ?o-projectivity of M is related to properties of ExtR (M, a?? R). Using results of S. Chase, S. Shelah and P. Eklof, the existence of non-projective D?1-projective modules is shown to undecidable, while both the Continuum Hypothesis and its denial (Plus Martin's Axiom) imply the existence of a reduced D?0-projective Z-module which is not free.
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