Generalization of $\mathcal{U}$-Generator and $M$-Subgenerator Related to Category $\sigma[M]$

Let $\mathcal{U}$ be a non-empty set of $R$-modules. $R$-module $N$ is generated by $\mathcal{U}$ if there is an epimorphism from $\oplus_{\Lambda}U_{\lambda}$ to $N$, where $U_{\lambda} \in \mathcal{U}$, for every $\lambda \in \Lambda$. $R$-module $M$ is a subgenerator for $N$ if $N$ is isomorphic to a submodule of an $M$-generated module. In this paper, we introduce a $\mathcal{U}_{V}$-generator, where $V$ be a submo\-dule of $\oplus_{\Lambda}U_{\lambda}$, as a generalization of $\mathcal{U}$-generator by using the concept of $V$-coexact sequence. We also provide a $\mathcal{U}_{V}$-subgenerator motivated by the concept of $M$-subgenerator. Furthermore, we give some properties of $\mathcal{U}_{V}$-generated and $\mathcal{U}_{V}$-subgenerated modules related to category $\sigma[M]$. We also investigate the existence of pullback and pushout of a pair of morphisms of $\mathcal{U}_{V}$-subgenerated modules. We prove that the collection of $\mathcal{U}_{V}$-subgenerated modules is closed under submodules and factor modules.