Weak diamond, weak projectivity, and transfinite extensions of simple artinian rings

We apply set-theoretic methods to study projective modules and their generalizations over transfinite extensions of simple artinian rings R. When R is hereditary and of cardinality at most $2^ω$, we prove that the Weak Diamond and CH imply that projectivity of an arbitrary module can be tested at the layer epimorphisms of R.

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