Semiprimary selfinjective algebras with at most countable dimensional   Jacobson quotient are QF
  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. 
Miodrag C. Iovanov
math.RA