Semiprimary selfinjective algebras with at most countable dimensional Jacobson quotient are QF

TL;DR

This paper proves that self-injective semiprimary algebras with at most countable dimensional Jacobson quotient are quasi-Frobenius, confirming a conjecture and providing simplified proofs for related cases.

Contribution

It confirms Faith's conjecture for a broad class of algebras and offers streamlined proofs for existing special cases.

Findings

Self-injective semiprimary algebras with countable Jacobson quotient are QF

Provides simplified proofs for known cases of Faith's conjecture

Extends the class of algebras known to be QF

Abstract

We give a positive answer to a conjecture of Faith stating that a self-injective semiprimary ring is QF, for algebras which are at most countable dimensional modulo their Jacobson radical. As a consequence of the method used, we also give short proofs of several other known positive cases of this conjecture.