= Injective hull of an Ore domain
{title2=$E({}_RR)=Q$}
For a domain satisfying the left <Ore condition>, its <division ring> of fractions $Q$ is an injective essential extension of the left regular module. For a homomorphism $f:I\to Q$ on a nonzero left ideal, choose $a\in I\setminus\{0\}$. A common left multiple $sx=ta$ gives $sf(x)=tf(a)$, hence $f(x)=xa^{-1}f(a)$; right multiplication by $a^{-1}f(a)$ extends it. The <Baer criterion> proves injectivity. Clearing a left denominator proves essentiality. The same calculation gives $\operatorname{End}_R({}_RQ)\cong Q^{\rm op}$, with the opposite ring required because right multiplications compose in reversed order.
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