nLab
restriction of scalars

Let $f:R\to S$ be a homomorphism of algebraic objects such as rings. Let ${\cdot }_S$ be an operation of $S$ on an object $M$ then by $r{\cdot }_{R}m:=f(r){\cdot }_{S}m$ is defined an action of $R$ on $M$.

It follows that we have a functor ${\rho }_f:\mathrm{SMod}\to \mathrm{RMod}$ sending ${\cdot }_S$ to ${\cot }_R$ which is a forgetful functor.

Adjointly we obtain a functor ${ϵ}_f:={\otimes }_{R}S:\mathrm{RMod}\to \mathrm{SMod}$ called the extension of scalars (see there for more) since for an $R$-module $M$ and the $R$-module $S$ we have that $M{\otimes }_{R}S$ is a well defined tensor product of $R$ modules which becomes an $S$ module by the operation of $S$ on itself in the second factor of the tensor. We have an adjunction $({ϵ}_f⊣{\rho }_f)$.