The rational hull of modules (Q6612115)

Let \(R\) be a unital ring and \(M\) a left \(R\)-module with injective hull \(E\). Let \(F=\{\vartheta\in \text{End}_R (E)\colon \vartheta (M)=0 \}\). The rational hull of \(M\) is \(\{x \in E \colon \forall \vartheta\in F,\ \vartheta(x)=0\}\) and \(M\) is rationally complete if it coincides with its rational hull.\N\NThe author reviews various known properties of the rational hull of a module and rationally complete modules, thereby obtaining new characterisations of the right ring of quotients of a module. They prove new properties of the rational hull, for example conditions for the rational hull of a direct sum to be the direct sum of rational hulls; for End\(_R(M)=\mathrm{End}_H(M)\), where \(H\) is the right ring of quotients of \(R\); and for the maximal right ring of quotients of the endomorphism ring of a module to be the endomorphism ring of the rational hull of the module.