Endochain conditions on modules and p-quasi-Frobenius rings

@misc{gera2026endochain,
  title = {{Endochain conditions on modules and $p$-quasi-Frobenius rings}},
  author = {Theophilus Gera and Amit Sharma and Manoj Kumar Patel and Pramod Kanwar},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.18492},
  eprint = {2610.18492},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.18492},
}

TY  - UNPB
TI  - Endochain conditions on modules and p-quasi-Frobenius rings
AU  - Theophilus Gera
AU  - Amit Sharma
AU  - Manoj Kumar Patel
AU  - Pramod Kanwar
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.18492
UR  - https://arxiv.org/abs/2610.18492
AB  - Analogous to quasi-Frobenius rings, dual rings, and Johns rings, the classes of p-quasi-Frobenius rings, p-dual rings, p-Johns rings are introduced and used to discuss and settle principal analogues of the Faith-Menal, Faith, and FGF conjectures. A chain symmetry theorem establishes that for quasi-injective semi-projective modules with semiprime endomorphism ring, endonoetherianity and endoartinianity are equivalent and force the endomorphism ring to be semisimple artinian. Within this unified setting, it is shown that (1) two-sided p-Johns rings are p-quasi-Frobenius, giving a principal Faith-Menal analogue, with a counterexample showing that the one-sided hypothesis is insufficient; (2) semiprimary principal right ideal rings that are right injective and right Kasch are p-quasi-Frobenius-this affirmatively answers a question of Faith (Q2) in the principal two-sided ideal ring case and settles the principal Faith conjecture, with a counterexample showing the principal right ideal ring to be indispensable; and (3) right p-Johns rings that are left CF and left perfect are p-quasi-Frobenius, yielding a principal FGF analogue, with a counterexample demonstrating that the hypotheses are essential. Wherever needed, examples are provided to show that various classes introduced are different from the existing classes in the literature.
ER  -