The Stacks project

Remark 35.4.27. It would make things easier to have a faithfully flat ring homomorphism $g: R \to T$ for which $T \to S \otimes _ R T$ has some extra structure. For instance, if one could ensure that $T \to S \otimes _ R T$ is split in $\textit{Rings}$, then it would follow that every property of a module or algebra which is stable under base extension and which descends along faithfully flat morphisms also descends along universally injective morphisms. An obvious guess would be to find $g$ for which $T$ is not only faithfully flat but also injective in $\text{Mod}_ R$, but even for $R = \mathbf{Z}$ no such homomorphism can exist.