Let $R$ be a PID and let $S \subseteq R$ by any multiplicatively closed set such that $0 \not \in S$. Assume that for any collection $\left\{s_{i}\right\}_{i \in \mathbb{N}} \subseteq S$ that $$\bigcap_{i \in \mathbb{N}} (s_{i}) \neq (0)$$
Is it possible for $R$ to have a nontrivial ideal or does $R$ have to be a field?