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. We prove that if R is small, then the Weak Diamond implies that projectivity of an arbitrary module can be tested at the layer epimorphisms of R.Comment: Revised version for J. Algebra, 11 pages. arXiv admin note: substantial text overlap with arXiv:2112.0964