Let $R$ be a commutative ring and $J(R)$ its Jacobson radical. It is easy to check that $J(R)$ consists of exactly those $f$ for which $1 + gf \in R^\times$ for all $g$.

Let $J'(R)$ be those $f$ for which $1 + uf \in R^\times$ for all units $u$. We might conjecture that in general $J(R) = J'(R)$. I assume this is false, since I can't find it anywhere, but I've been unable to come up with a counterexample - what are some counterexamples? Is there a nice characterization of when $J(R) = J'(R)$ holds?