New Characterizations of pseudo-Frobenius rings and a generalization of the FGF conjecture

TL;DR

This paper introduces new characterizations of pseudo-Frobenius and quasi-Frobenius rings using tight modules, and proposes extensions of the FGF and CF conjectures linking them to classical PF ring theory.

Contribution

It offers novel characterizations of pseudo-Frobenius rings and extends the FGF and CF conjectures, connecting them with classical PF ring theory using set-theoretic methods.

Findings

New characterizations of pseudo-Frobenius rings

Extensions of FGF and CF conjectures proposed

Set-theoretic counting techniques applied

Abstract

We provide new characterizations of pseudo-Frobenius and quasi-Frobenius rings in terms of tight modules. In the process, we also provide fresh perspectives on FGF and CF conjectures. In particular, we propose new natural extensions of these conjectures which connect them with the classical theory of PF rings. Our techniques are mainly based on set-theoretic counting arguments initiated by Osofsky. Several corollaries and examples to illustrate their applications are given.