Injective hull of an Ore domain ($E({}_{R}R)=Q$)

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 $f(x)=x{a}^{-1}f(a)$; right multiplication by $a^{-1}f(a)$ extends it. The Baer criterion proves injectivity. Clearing a left denominator proves essentiality. The same calculation gives ${\mathrm{End}}_R({}_{R}Q)\cong Q^{\mathrm{op}}$, with the opposite ring required because right multiplications compose in reversed order.