Completely integrally closed modules and rings

A ring $A$ is a completely integrally closed right $A$-module if and only if the maximal right ring of quotients $Q_\mathrm{max}(A)$ of $A$ is an injective right $A$-module and $A$ is a right completely integrally closed subring in $Q_\mathrm{max}(A)$. A right Noetherian, right integrally closed ring $A$ is a completely integrally closed right $A$-module.