Consider a morphism of commutative rings $h\colon R\rightarrow S$. This yields the two functors $h_*\colon{\sf Mod}(S)\rightarrow{\sf Mod}(R)$ (scalar restriction) and $h^*\colon{\sf Mod}(R)\rightarrow{\sf Mod}(S)$ (scalar extension), and $h^*$ is left adjoint to $h_*$. The unit of this adjunction is for an $R$-module $M$ given by the morphism $$\rho_h(M)\colon M\rightarrow h_*(h^*(M)),\;x\mapsto x\otimes 1_S.$$
It is easy to see the following:
a) $\rho_h$ is an isomorphism if and only if $h$ is bijective;
b) $\rho_h$ is an epimorphism if and only if $h$ is surjective;
c) If $\rho_h$ is a monomorphism then $h$ is injective.
It is furthermore clear that the converse of c) need not be true. So, we may ask the following question:
For which (necessarily injective) morphisms of rings $h\colon R\rightarrow S$ is the unit $\rho_h$ of the adjunction $(h^*,h_*)$ a monomorphism?
An example of a class of such morphisms are those whose source is absolutely flat.
