  Recently, the rings whose injective right modules are R-projective
(respectively, max-projective) were investigated and studied in [2]. Such ring
are called right almost-QF (respectively, max-QF). In this paper, our aim is to
give some further characterization of these rings over more general classes of
rings, and address several questions about these rings. We obtain
characterizations of max-QF rings over several classes of rings including
local, semilocal right semihereditary, right nonsingular right noetherian and
right nonsingular right finite dimensional rings. We prove that for a ring R
being right almost-QF and right max-QF are not left-right symmetric. We also
show that right almost-QF and right max-QF rings are not closed under factor
rings. This leads to consider the rings all of whose factor rings are almost-QF
and max-QF.
