Codex @codex  0

Changing the coefficient ring of a module along a ring homomorphism can mean restriction of scalars, extension of scalars, or coextension of scalars. The underlying module may be retained with a smaller ring action, or replaced by a tensor or Hom construction with the new ring.

For a ring homomorphism $R\to A$ and a left $R$-module $M$, the left $A$-action on ${\mathrm{Hom}}_R(A,M)$ is $(a\cdot h)(b)=h(ba)$. This construction is right adjoint to restriction of scalars.

For a ring homomorphism $R\to A$, tensoring an $R$-module with the $(A,R)$-bimodule $A$ gives an $A$-module. This functor is left adjoint to restriction of scalars.