Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.18492

Endochain Conditions on Modules and $p$-quasi-Frobenius Rings

Theophilus Gera·Amit Sharma·Manoj Kumar Patel·Pramod Kanwar
View original source ↗Read source document ↗Export metadata ↓

Abstract

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 principally 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 hypotheses 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.

Explore related subjects

math.RA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Theophilus Gera, Amit Sharma, Manoj Kumar Patel, Pramod Kanwar. 2026-10-13. Endochain Conditions on Modules and $p$-quasi-Frobenius Rings. https://arxiv.org/abs/2610.18492

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Orthogonality of Defect Ideals for Jordan Homomorphisms

Let $φ\colon A\to B$ be a Jordan homomorphism of associative algebras over a field $\mathbb{F}$ of characteristic different from $2$, with defect ideals $V_φ$ and $W_φ$. If $A=\operatorname{Alg}([[A,A],[A,A]])$, then $V_φW_φ=W_φV_φ=0$ and $V_φ\cap W_φ\subseteq\operatorname{Ann}(\operatorname{Alg}(φ(A)))$. Moreover, the induced map modulo $V_φ\cap W_φ$ is standard. The annihilator bound is sharp, even in finite dimension. As an application, when $A$ is unital and $A=K(A)$, every Jordan homomorphism defined on $A$ is splittable and standard. In this setting, the corresponding result of Brešar and Zelmanov holds without the surjectivity hypothesis.

math.RA↗
arXiv2026

On the Real Waring Rank of a Biharmonic Form

We develop an effective method for determining the real Waring rank and the real border rank of biharmonic binary forms. By means of real apolarity, we convert both of these problems into membership questions for regions bounded by a hypocycloid and an epicycloid. We further provide a computable algebraic criterion for the relevant memberships, which involves only the signs of certain real-valued resultants.

math.RA↗
arXiv2026

Actions of central Morita monoidal groupoids, and seven term exact sequences of all types

We introduce a strict monoidal groupoid associated with a given commutative ring $K$. It acts on the Clifford extension systems of $K$-central algebras which are graded by a group $G$, as well as on the generalized collective characters, both of type $ϕ\in$ Hom$(G,$ Aut$(K))$. These actions partition the two sets above with respect to graded Morita equivalence. By that we establish a seven term exact sequence of type $ϕ$, which boils down to the celebrated Chase-Harrison-Rosenberg sequence when $ϕ$ is Galois.

math.RA↗