Given any algebra $R,$ when does the forgetful functor
$R\text{-}Mod \rightarrow Vec$
have a *right* adjoint?
Does this imply any finiteness conditions on R?
Is there a book/paper discussing this?

I've assumed $R$ is $k$ algebra where $k$ is a field. but if $k$ is not a field, and just a commutative ring then Marco's answer should hold up still with replacing $Vec$ by $k-Mod$.