On the rings whose injective right modules are max-projective

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.