Consider a morphism of commutative rings $h\colon R\rightarrow S$. This gives rise to a functor $h^*\colon{\sf Mod}(R)\rightarrow{\sf Mod}(S)$, called scalar extension by means of $h$. This functor has a right adjoint, hence it commutes with inductive limits. We may ask now whether or not $h^*$ commutes with *projective* limits.

Clearly, $h^*$ is left exact if and only if $h$ is flat. Therefore (and using some general nonsense), $h^*$ commutes with projective limits if and only if $h$ is flat and $h^*$ commutes with infinite products. Flatness of $h$ does not imply that $h^*$ commutes with infinite products. So, the question is as follows:

Are there some conditions on a morphism of rings $h\colon R\rightarrow S$ that ensure that the scalar extension functor $h^*$ commutes with infinite products?

Lectures on modules and rings,Proposition 4.44.) $\endgroup$ – Fred Rohrer Mar 25 '15 at 18:56