A -D3 Modules and A -D4 Modules

Let $\mathcal{A}$ 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 $\mathcal{A}$ -D3 if, whenever $N$ and $K$ are direct summands of $M$ with $M=N+K$ and $M/K\in \mathcal{A}$ , then $N\cap K$ is also a direct summand of $M$ ; $M$ is called an $\mathcal{A}$ -D4 module, if whenever $M=B\oplus A$ where $B$ and $A$ are submodules of $M$ and $A\in \mathcal{A}$ , 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.