Given any ring homomorphism $R \to S$, if $S$ is a flat right $R$-module, then any injective left $S$-module is also injective as a left $R$-module.
Now, I'm wondering whether the converse is true. That is, if any injective left $S$-module is also injective as a left $R$-module, does it imply that $S$ must then be a flat right $R$-module?
Some evidence includes the following:
- If all left $R$-modules are injective, then $R$ is semisimple and the same must also be true for all right $R$-modules, which also implies that all right or left $R$-modules are flat (in fact, projective).
- Suppose that $R=\mathbb{Z}$ (or any Dedekind domain) and all left $S$-modules are divisible (equivalently, injective) $R$-modules (divisible abelian groups in the case $R=\mathbb{Z}$).
- Then, any nonzero element of $R$ is mapped to a unit in $S$, which then cannot be a zero divisor. This implies that $S$ must be a torsion-free (equivalently, flat) $R$-module.