A ring R is called a (left) \(\pi \) -V-ring if, for every simple (left) R-module M, the injective hull E(M) of M is of finite length. The main goal of this paper is to introduce and study the notion of a (left) \(\pi \) -V-module. First, we define a radical \(G^R\) on the category \({{\,\textrm{Mod}\,}}(R)\) of R-modules. An R-module is called a \(\pi \) -V-module if \(G^R(N)=0\) for any module \(N\in \sigma [M]\) . We prove that a (left) module M over a ring R is a \(\pi \) -V-module if and only if M is co-noetherian and co-artinian. As an application, we deduce that a ring R is a (left) \(\pi \) -V-ring if and only if R is a (left) co-noetherian and co-artinian ring.