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

TL;DR

This paper explores how set-theoretic principles like the Weak Diamond and CH influence the projectivity of modules over transfinite extensions of simple artinian rings, providing new insights into module theory under these conditions.

Contribution

It demonstrates that under certain set-theoretic assumptions, projectivity can be tested at specific layer epimorphisms for transfinite extensions of simple artinian rings.

Findings

Weak Diamond and CH imply projectivity testing at layer epimorphisms

Results apply to hereditary rings of size at most 2^ω

Advances understanding of module projectivity in set-theoretic context

Abstract

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 $2^\omega$, we prove that the Weak Diamond and CH imply that projectivity of an arbitrary module can be tested at the layer epimorphisms of R.