On two conjectures of Faith

TL;DR

This paper proves two of Faith's conjectures for profinite algebras, showing that certain conditions imply the algebra is finite dimensional and quasi-Frobenius, and explores related ring-theoretic properties.

Contribution

It establishes the validity of Faith's conjectures in the setting of profinite algebras and introduces a dual notion of PF rings for pseudocompact algebras.

Findings

Profinite semiartinian selfinjective algebras are finite dimensional and QF.

FGF profinite algebras are finite dimensional and QF.

A new dual notion of PF rings for pseudocompact algebras is introduced.

Abstract

We prove that a profinite algebra whose left (right) cyclic modules are torsionless is finite dimensional and QF. We give a relative version of the notion of left (right) PF ring for pseudocompact algebras and prove it is left-right symmetric and dual to the notion of quasi-co-Frobenius coalgebras. We also prove two ring theoretic conjectures of Faith, in the setting (and supplementary hypothesis) of profinite algebras: any profinite semiartinian selfinjective algebra is finite dimensional and QF, and any FGF profinite algebra is finite dimensional QF.