$\mathcal{A}$-D3 Modules and $\mathcal{A}$-D4 Modules

Abstract

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.