Let $R$ be an integral domain. Is the $R$-Module $\DeclareMathOperator{\Quot}{Quot}\Quot(R)/R$ injective? ($\Quot R$ denotes the fraction field of $R$.)
By Baer's criterion it is sufficient to show that for any ideal $J \trianglelefteq R$, all $R$-linear maps $J \to \Quot(R)$ are of the form $a \mapsto c a$ for some $c \in R$.
The statement is true if $R$ is a PID because $\Quot(R)/R$ is a divisible $R$-Module.