For any given radical $\rho$, the $\rho$-rational extensions of modules is a stronger version of rational extensions of modules, and its properties are studied. However, when $\rho$ is the identity functor both these concepts coincide. In most of the work of this thesis, the ring of the module structure is not assumed to be commutative. Just as every module has a rational completion, it is shown that for any given radical $\rho$, every module has a $\rho$-rational completion, which is unique upto isomorphism in its $\rho$-divisible hull. For any ring R with identity, the $\rho$-rational completion $\overline{R} \sp\rho$ of R is a ring. The $\rho$-rational completion of a module in terms of filters are studied, and the behavior of $\rho$-rationally complete modules under the formation of direct products and direct sums is obtained. Finally, we establish the invariance of the $\rho$-rational completeness of a module under a change of rings