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The aim of this paper is to study the notions of $\mathcal{A}$-C3 and$\mathcal{A}$-D3 modules for some class $\mathcal{A}$ of right modules. Severalcharacterizations of these modules are provided and used to describe somewell-known classes of rings and modules. For example, a regular right$R$-module $F$ is a $V$-module if and only if every $F$-cyclic module $M$ is an$\mathcal{A}$-C3 module where $\mathcal{A}$ is the class of all simplesubmodules of $M$. Moreover, let $R$ be a right artinian ring and$\mathcal{A}$, a class of right $R$-modules with local endomorphisms,containing all simple right $R$-modules and closed under isomorphisms. If allright $R$-modules are $\mathcal{A}$-injective, then $R$ is a serial artinianring with $J^{2}(R)=0$ if and only if every $\mathcal{A}$-C3 right $R$-moduleis quasi-injective, if and only if every $\mathcal{A}$-C3 right $R$-module isC3.

On classes of C3 and D3 modules