Faith's problem on R-projectivity is undecidable

TL;DR

This paper demonstrates that the question of whether right testing rings are necessarily right perfect is undecidable within ZFC, using consistency results and examples related to Faith's problem on R-projectivity.

Contribution

It proves the consistency of the existence of right testing rings that are not right perfect within ZFC, showing the problem's undecidability and providing examples for small modules.

Findings

The existence of right testing, non-right perfect rings is consistent with ZFC.

Faith's problem on R-projectivity is undecidable in ZFC.

Examples of non-right perfect rings where the Dual Baer Criterion holds for small modules.

Abstract

In \cite{F}, Faith asked for what rings $R$ does the Dual Baer Criterion hold in Mod-$R$, that is, when does $R$-projectivity imply projectivity for all right $R$-modules? Such rings $R$ were called right testing. Sandomierski proved that if $R$ is right perfect, then $R$ is right testing. Puninski et al.\ \cite{AIPY} have recently shown for a number of non-right perfect rings that they are not right testing, and noticed that \cite{T2} proved consistency with ZFC of the statement {\lq}each right testing ring is right perfect{\rq} (the proof used Shelah's uniformization). Here, we prove the complementing consistency result: the existence of a right testing, but not right perfect ring is also consistent with ZFC (our proof uses Jensen-functions). Thus the answer to the Faith's question above is undecidable in ZFC. We also provide examples of non-right perfect rings such that the Dual Baer Criterion holds for {\lq}small{\rq} modules (where {\lq}small{\rq} means countably generated, or $\leq 2^{\aleph_0}$-presented of projective dimension $\leq 1$).