A-D3 Modules and A-D4 Modules

Let A be a class of some right R-modules that is closed under isomorphisms, and let M be a right R-module. Then M is called A-D3 if, whenever N and K are direct summands of M with M=N+K and M/K∈A, then N∩K is also a direct summand of M; M is called an A-D4 module, if whenever M=B⊕A where B and A are submodules of M and A∈A, then every epimorphism f:B⟶A splits. Several characterizations and properties of these classes of modules are investigated. As applications, some new characterizations of semisimple Artinian rings, quasi-Frobenius rings, von Neumann regular rings, semiregular rings, perfect rings, semiperfect rings, hereditary rings, semihereditary rings, and PP rings are given