$(m,n)$-$C2$ modules and $(m,n)$-$D2$ modules

We study the concept of $(m,n)$-$C2$ modules with $m,n$ positive integers, which unifies strongly $C2$, $n$-$C2$ and $GC2$ modules. Several characterizations are obtained. It is shown that $R^{(\mathbb{N})}$ is $(m,n)$-$C2$ as a right $R$-module if and only if $R$ is right perfect and right strongly $C2$. Connections between an $(m,n)$-$C2$ module and its endomorphism ring are also studied. We prove that if the endomorphism ring of an $R$-module $M$ is a right $(m,n)$-$C2$ ring, then $M$ is an $(m,n)$-$C2$ module. Also we obtain some dual statements of $(m,n)$-$D2$ modules. Some characterizations of (semi)perfect and (semi)regular rings are studied. We show that $S=End(M_R)$ is a regular ring if and only if $M$ is a dual Rickart module and $(m,n)$-$D2$ with $m> n$.