The rational hull of modules

In this paper, we provide several new characterizations of the maximal right ring of quotients of a ring by using the relatively dense property. As a ring is embedded in its maximal right ring of quotients, we show that the endomorphism ring of a module is embedded into that of the rational hull of the module. In particular, we obtain new characterizations of rationally complete modules. The equivalent condition for the rational hull of the direct sum of modules to be the direct sum of the rational hulls of those modules under certain assumption is presented. For a right $H$-module $M$ where $H$ is a right ring of quotients of a ring $R$, we provide a sufficient condition to be $\text{End}_R(M)=\text{End}_H(M)$. Also, we give a condition for the maximal right ring of quotients of the endomorphism ring of a module to be the endomorphism ring of the rational hull of a module.