Rabinowitch's trick

Statement

Let $R$ be a commutative unital ring. The following are equivalent:

1. $R$ is a Jacobson ring
2. If $P$ is a prime ideal of $R$ and if $S:=R/P$ contains an element $b\ne 0$ such that $S[b^{-1}]$ is a field, then $S$ is a field