Let $R$ be a a commutative ring with a unit element, then one can associate to $R$ a Boolean ring $B(R)$, as in this text by Bergman, last line of page 594. (I guess this is a very classical thing. Explicitly, $B(R)$ is the set of idempotent elements of $R$ with the operations $e \oplus f = e + f - 2ef$ and multiplication inherited from $R$.) Bergman shows that the functor $R \mapsto B(R)$ has a left adjoint - the one you would expect. I suspect that it does not have a right adjoint, but I don't see how to prove this (if it is true).
All ideas would be welcome!