AbstractA moduleMis called aD4-module if, wheneverAandBare submodules ofMwithM=A⊕Bandf:A→Bis a homomorphism with Imfa direct summand ofB, then Kerfis a direct summand ofA. The class ofD4-modules contains the class ofD3-modules, and hence the class of semi-projective modules, and so the class of Rickart modules. In this paper we prove that, over a commutative Dedekind domainR, for anR-moduleMwhich is a direct sum of cyclic submodules,Mis direct projective (equivalently, it is semi-projective) iffMisD3 iffMisD4. Also we prove that, over a prime PI-ring, for a divisibleR-moduleX,Xis direct projective (equivalently, it is Rickart) iffX⊕XisD4. We determine someD3-modules andD4-modules over a discrete valuation ring, as well. We give some relevant examples. We also provide several examples onD3-modules andD4-modules via quivers.