The rational hull of modules

TL;DR

This paper explores new characterizations of the maximal right ring of quotients and rational hulls of modules, providing insights into their structure and relationships with endomorphism rings.

Contribution

It introduces novel characterizations of the maximal right ring of quotients and rationally complete modules, and examines conditions relating endomorphism rings to rational hulls.

Findings

New characterizations of the maximal right ring of quotients

Conditions for the rational hull of a direct sum to be the sum of rational hulls

Criteria for endomorphism rings of modules over different rings

Abstract

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.