max-projective modules

Abstract

A right R-module M is called max-projective provided that each homomorphism f : M → R/I where I is any maximal right ideal, factors through the canonical projection π : R → R/I. We call a ring R right almost-QF (resp. right max-QF ) if every injective right R-module is R-projective (resp. max-projective). This paper attempts to understand the class of right almost-QF (resp. right max-QF ) rings. Among other results, we prove that a right Hereditary right Noetherian ring R is right almost-QF if and only if R is right max-QF if and only if R = S × T , where S is semisimple Artinian and T is right small. A right Hereditary ring is max-QF if and only if every injective simple right R-module is projective. Furthermore, a commutative Noetherian ring R is almost-QF if and only if R is max-QF if and only if R = A × B, where A is QF and B is a small ring.