# The Dual Baer Criterion for non-perfect rings

## TL;DR

This paper investigates the validity of the Dual Baer Criterion (DBC) for non-perfect rings, revealing that for certain classes of rings, DBC either fails or is independent of ZFC, expanding understanding of module theory.

## Contribution

It demonstrates that all small semiartinian von Neumann regular rings with artinian primitive factors belong to the class where DBC is independent of ZFC.

## Key findings

- DBC fails in ZFC for some non-right perfect rings.
- DBC is independent of ZFC for a broad class of rings.
- Includes new examples of rings where DBC's status is undecidable.

## Abstract

Baer's Criterion for Injectivity is a basic tool of the theory of modules and complexes of modules. Its dual version (DBC) is known to hold for all right perfect rings, but its validity for non-right perfect rings is a complex problem (first formulated by Faith in 1976 \cite{F}). Recently, it has turned out that there are two classes of non-right perfect rings: 1. those for which DBC fails in ZFC, and 2. those for which DBC is independent of ZFC. First examples of rings in the latter class were constructed in \cite{T4}; here, we show that this class contains all small semiartinian von Neumann regular rings with primitive factors artinian.