ac.commutative algebra - Primes in a (commutative) Jacobson ring

The result is true in general.

We may assume a counterexample is given in the form of a domain $R$ satisfying the second property but with nontrivial Jacobson radical, i.e. the closed points of Spec $R$ are not dense. Let D be an affine open neighborhood of $(0)$ in Spec $R$ which contains no closed points. Since D is affine, there exists some $xin$ D which is closed in D. That is,

$overline{lbrace xrbrace}setminus xsubsettext{Spec }Rsetminus D$.

Since $D$ is open, this implies

$xnotin overline{overline{lbrace xrbrace}setminus x}$,

but this contradicts the requirements of the second property.