Proposition 4

Let $A$ be a principal ideal domain and $(\mathbf{p}_\lambda)_{\lambda \in L}$ a representative system of extremal elements of $A$ (Algebra*, Chapter VII, § 1, no. 3, Definition 2). For $A$ to be a Jacobson ring, it is necessary and sufficient that $L$ be infinite.*

The maximal ideals of $A$ are the $A\mathbf{p}_\lambda$ (loc. cit., no. 2, Proposition 2). If $L$ is finite, their intersection is the ideal $Ax$, where $x = \prod_{\lambda \in L} \mathbf{p}_\lambda$ (ibid.) and hence different from $(0)$; on the other hand, if $L$ is infinite, the intersection of the $A\mathbf{p}_\lambda$ is $(0)$, every element $\neq 0$ of $A$ being divisible by only a finite number of extremal elements (loc. cit., no. 3, Theorem 2). The proposition then follows from the fact that $(0)$ is the only prime ideal which is not maximal in $A$ (Algebra, Chapter VI, § 1, no. 13, Proposition 14 (DIV)).