A left R-module M is called max-injective (or m-injective for short) if for any maximal left ideal I, any homomorphism f : I → M can be extended to g : R → M , if and only if Ext 1 R (R/I, M) = 0 for any maximal left ideal I. A left R-module M is called max-projective (or m-projective for short) if Ext 1 R (M, N) = 0 for any max-injective left R-module N. We prove that every left R-module has a special m-projective precover and a special m-injective preenvelope. We characterize C-rings, SF rings and max-hereditary rings using m-projective and m-injective modules.

Weakening the notion of [Formula: see text]-projectivity, a right [Formula: see text]-module [Formula: see text] is called max-projective provided that each homomorphism [Formula: see text], where [Formula: see text] is any maximal right ideal, factors through the canonical projection [Formula: see text]. We study and investigate properties of max-projective modules. Several classes of rings whose injective modules are [Formula: see text]-projective (respectively, max-projective) are characterized. For a commutative Noetherian ring [Formula: see text], we prove that injective modules are [Formula: see text]-projective if and only if [Formula: see text], where [Formula: see text] is [Formula: see text] and [Formula: see text] is a small ring. If [Formula: see text] is right hereditary and right Noetherian then, injective right modules are max-projective if and only if [Formula: see text], where [Formula: see text] is a semisimple Artinian and [Formula: see text] is a right small ring. If [Formula: see text] is right hereditary then, injective right modules are max-projective if and only if each injective simple right module is projective. Over a right perfect ring max-projective modules are projective. We discuss the existence of non-perfect rings whose max-projective right modules are projective.

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