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?

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 $k-Mod$.

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

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?

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

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