On a subclass of the class of π-projective modules

Let [Formula: see text] be a commutative ring. We call an [Formula: see text]-module [Formula: see text] s-[Formula: see text]-projective if for any submodules [Formula: see text] and [Formula: see text] of [Formula: see text] with [Formula: see text], there exists [Formula: see text] such that [Formula: see text] and [Formula: see text]. We describe the structure of direct summands and maximal submodules of an s-[Formula: see text]-projective module. We show that every s-[Formula: see text]-projective module is Abelian (that is, every direct summand is fully invariant). We characterize when a finite direct sum of s-[Formula: see text]-projective modules is s-[Formula: see text]-projective. We describe the structure of s-[Formula: see text]-projective modules over perfect rings and Dedekind domains. Also, Bass rings and semiperfect rings are characterized using s-[Formula: see text]-projectivity.